Searching for New Integrals in the Euler–Poisson Equations

Abstract

In the classical problem of the motion of a rigid body around a fixed point, which is described by the Euler–Poisson equations, we propose a new method for computing cases of integrability: first, we provide algorithms for computing values of parameters ensuring potential integrability, and then we select cases of global integrability. By this method we have obtained all the known cases of global integrability and six new cases of potential integrability for which the absence of their global integrability is proven.

1. Introduction

The study of the motion of a rigid body with a fixed point has long been recognized as a cornerstone of classical mechanics. From a theoretical perspective, this problem provides an archetypal setting for investigating rotational motion, analyzing dynamical invariants, and testing methods of integrability. Its significance is enhanced by the profound interplay between geometry and analysis, which emerges from the underlying equations. Consequently, the search for integrable cases in which there is a sufficient number of independent first integrals in involution remains a subject of continuing interest. These cases serve as illuminating examples of completely integrable systems within finite-dimensional Hamiltonian mechanics, with far-reaching applications in both mathematical theory and concrete physical models.

Historically, the quest for integrable cases of rigid body motion goes back to seminal contributions by Euler [1] and Lagrange [2], who laid out the fundamental equations of motion for a rigid body about a fixed point. The classic Kovalevskaya result [3,4] demonstrated a highly non-trivial integrable configuration under special symmetry conditions, which stimulates further efforts to uncover additional solutions. Subsequent advances by Grioli, Chaplygin, Yehia, and others revealed new algebraic and geometric structures underlying integrability. More recently [5], the geometric and topological classification of integrable or partially integrable cases has been explored through the lens of symplectic and Poisson geometry.

A diverse array of methodologies has been utilized in the pursuit of these integrable configurations. Traditional approaches frequently employ the assumption of polynomial forms of integrals, with subsequent verification of closure under Poisson brackets, occasionally invoking symmetries and the corresponding conservation laws as elucidated by Noether’s theorem. Beyond these classical methods, contemporary techniques such as the Ziglin–Morales–Ramis differential Galois approach have demonstrated substantial efficacy in identifying obstructions to integrability. In addition, advanced algebraic and topological instruments have allowed the classification of existing integrable families. Recent scholarly reviews offer extensive overviews of these methodologies and the evolving perspectives surrounding this issue (see, for example, the survey [6]), in which the authors examine both traditional and contemporary techniques, delineate prominent integrable systems, and articulate open questions that continue to propel advancements in the field.

This study introduces an innovative approach to identify integrable cases of the rigid top with a fixed point. By using the structural characteristics of the dynamical equations and developing new invariants within a unified framework, we seek to advance previous methodologies and expand the repertoire of recognized integrable configurations.

The system of Euler–Poisson equations (1750) (or, in short, EP equations) is a real autonomous system of six ordinary differential equations (ODEs).

$$\begin{array}{cc}\hfill & A{p}^{\prime }+(C-B)qr=Mg\left(y_0{\gamma }_3-z_0{\gamma }_2\right),\hfill \\ \hfill & B{q}^{\prime }+(A-C)pr=Mg\left(z_0{\gamma }_1-x_0{\gamma }_3\right),\hfill \\ \hfill & C{r}^{\prime }+(B-A)pq=Mg\left(x_0{\gamma }_2-y_0{\gamma }_1\right),\hfill \\ \hfill & {\gamma }_1^{\prime }=r{\gamma }_2-q{\gamma }_3,{\gamma }_2^{\prime }=p{\gamma }_3-r{\gamma }_1,{\gamma }_3^{\prime }=q{\gamma }_1-p{\gamma }_2,\hfill \end{array}$$

(1)

with dependent variables $p,q,r,{\gamma }_1,{\gamma }_2,{\gamma }_3$ and parameters $A,B,C$, $x_0,y_0,z_0$, which satisfy the triangle inequalities

$$0<A⩽B+C,0<B⩽A+C,0<C⩽A+B.$$

(2)

Here, the prime indicates differentiation over the independent variable time t, $Mg$ is the weight of the body, $A,B,C$ are the principal moments of inertia of the rigid body, $x_0,y_0,z_0$ are the coordinates of the center of mass of the rigid body, and ${\gamma }_1,{\gamma }_2,{\gamma }_3$ are the vertical directional cosines.

EP equations describe the motion of a rigid top around a fixed point [7] and have the following three first integrals:

$$\begin{array}{cc}\hfill \mathit{energy}:& I_1 \stackrel{\mathrm{def}}{=} A{p}^2+B{q}^2+C{r}^2-2Mg\left(x_0{\gamma }_1+y_0{\gamma }_2+z_0{\gamma }_3\right)=h=\mathrm{const},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathit{momentum}:& I_2 \stackrel{\mathrm{def}}{=} Ap{\gamma }_1+Bq{\gamma }_2+Cr{\gamma }_3=l=\mathrm{const},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathit{geometric}:& I_3 \stackrel{\mathrm{def}}{=} {\gamma }_1^2+{\gamma }_2^2+{\gamma }_3^2=1.\hfill \end{array}$$

(5)

EP equations are integrable if there is a fourth general integral $I_4$. So far, four cases of integrability are known:

Case 1. 

Euler-Poinsot: $x_0=y_0=z_0=0$ and the fourth integral is

$$I_4 \stackrel{\mathrm{def}}{=} A^2{p}^2+B^2{q}^2+C^2{r}^2=\mathrm{const}.$$

(6)

Case 2. 

Lagrange-Poisson: $B=C$, $x_0\ne 0$, $y_0=z_0=0$, and the fourth integral is

$$I_4 \stackrel{\mathrm{def}}{=} p=\mathrm{const}.$$

(7)

Case 3. 

Kovalevskaya (1890): $A=B=2C$, $x_0\ne 0$, $y_0=z_0=0$, and

$$I_4 \stackrel{\mathrm{def}}{=} {\left(p^2-q^2+c{\gamma }_1\right)}^2+{\left(2pq+c{\gamma }_2\right)}^2=\mathrm{const},$$

(8)

where $c=Mg{x}_0/C$.

Case 4. 

Kinematic symmetry: $A=B=C$ and $I_4 \stackrel{\mathrm{def}}{=} x_{0}p+y_{0}q+z_{0}r=\mathrm{const}$. It is derived from Case 2.

Case 5. 

Bruno–Batkhin (2024) [8]: $A=B=2C$, $x_0\ne 0$, $y_0\ne 0$, $z_0=0$, the fourth integral is

$$I_4 \stackrel{\mathrm{def}}{=} {\left(p^2-q^2+c{\gamma }_1-d{\gamma }_2\right)}^2+{\left(2pq+d{\gamma }_1+c{\gamma }_2\right)}^2=\mathrm{const},$$

where $c=Mg{x}_0/C$, $d=Mg{y}_0/C$.

Below ℓ is the number of non-zero values among $x_0,y_0,z_0$.

The structure of the paper is the following. In Section 2 we introduce the permutation group that provides automorphisms of the EP equations. Section 3 is devoted to the description of the theoretical background of the applied methods, and the next Section 4 details some peculiarities of the computational process. In Section 5 the description of the families of stationary points (SP) of the EP equations is given. Section 6, Section 7, Section 8 and Section 9 contain the details of computations, respectively, for ${\displaystyle \mathsf{\ell }}=0,1,2,3$ based on the theoretical approach and its implementations given in the previous Sections. And the final Section 10 summarizes the results currently obtained: we have obtained all the known cases of global integrability and six new cases of potential integrability for which the absence of their global integrability is proven.

2. Symmetries

EP equations allow for a permutation symmetry group. Let us write the variables and parameters of the system (1) as a matrix

$$N_1=\left(\begin{array}{ccc}p& q& r\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\\ A& B& C\\ x_0& y_0& z_0\end{array}\right)$$

(9)

If we rearrange two columns of the $N_1$ matrix, we obtain the original system with a different order of equations and opposite sign in the first derivative.

For example, when rearranging the first and second columns of the matrix (9), i.e., substituting

$$q\leftrightarrow p,{\gamma }_2\leftrightarrow {\gamma }_1,B\leftrightarrow A,y_0\leftrightarrow x_0,t\to -t,$$

(10)

the third and sixth equations of the system (1) go into themselves, and the first and second equations, as well as the fourth and fifth equations, are reversed:

$$\begin{array}{cc}\hfill & B{q}^{\prime }+(A-C)pr=Mg\left(z_0{\gamma }_1-x_0{\gamma }_3\right),\hfill \\ \hfill & A{p}^{\prime }+(C-B)qr=Mg\left(y_0{\gamma }_3-z_0{\gamma }_2\right),\hfill \\ \hfill & C{r}^{\prime }+(B-A)pq=Mg\left(x_0{\gamma }_2-y_0{\gamma }_1\right),\hfill \\ \hfill & {\gamma }_3^{\prime }=q{\gamma }_1-p{\gamma }_2,{\gamma }_1^{\prime }=r{\gamma }_2-q{\gamma }_3,{\gamma }_2^{\prime }=p{\gamma }_3-r{\gamma }_1.\hfill \end{array}$$

(11)

This arrangement of three columns has its own matrix

$$N_2=\left(\begin{array}{ccc}q& p& r\\ {\gamma }_2& {\gamma }_1& {\gamma }_3\\ B& A& C\\ y_0& x_0& z_0\end{array}\right).$$

(12)

There are in total six variants of arrangements of three columns, and to each of them corresponds its own matrix $N_k$, $k=1,\dots ,6$, namely the matrices (9), (12) and the following matrices

$$\begin{array}{c}{N}_3=\left(\begin{array}{ccc}p& r& q\\ {\gamma }_1& {\gamma }_3& {\gamma }_2\\ A& C& B\\ x_0& z_0& y_0\end{array}\right),N_4=\left(\begin{array}{ccc}q& r& p\\ {\gamma }_2& {\gamma }_3& {\gamma }_1\\ B& C& A\\ y_0& z_0& x_0\end{array}\right),N_5=\left(\begin{array}{ccc}r& q& p\\ {\gamma }_3& {\gamma }_2& {\gamma }_1\\ C& B& A\\ z_0& y_0& x_0\end{array}\right),N_6=\left(\begin{array}{ccc}r& p& q\\ {\gamma }_3& {\gamma }_1& {\gamma }_2\\ C& A& B\\ z_0& x_0& y_0\end{array}\right).\end{array}$$

(13)

Insofar as an even permutation is a composition of two elementary permutations, in this case, the sign of the independent variable t is not changed.

So, the matrix $N_3$ defines the substitution

$$q\leftrightarrow r,{\gamma }_2\leftrightarrow {\gamma }_3,B\leftrightarrow C,y_0\leftrightarrow z_0,t\to -t.$$

(14)

The matrix $N_4$ defines the cyclic substitution

$$p\to q\to r\to p,{\gamma }_1\to {\gamma }_2\to {\gamma }_3\to {\gamma }_1,A\to B\to C\to A,x_0\to y_0\to z_0\to x_0.$$

(15)

The matrix $N_5$ defines the substitution

$$p\leftrightarrow r,{\gamma }_1\leftrightarrow {\gamma }_3,A\leftrightarrow C,x_0\leftrightarrow z_0,t\to -t.$$

(16)

The matrix $N_6$ defines the cyclic substitution

$$p\to r\to q\to p,{\gamma }_1\to {\gamma }_3\to {\gamma }_2\to {\gamma }_1,A\to C\to B\to A,x_0\to z_0\to y_0\to x_0.$$

(17)

3. Theory

3.1. Local and Global Integrability

In [9] the notion local integrability was introduced:

The ODE system is locally integrable near a stationary point of the system if it has enough analytic integrals in the vicinity of the SP. It is evident that an integrable system is locally integrable at each of its stationary points.

Hypothesis 1

(Edneral [10]). If an autonomous polynomial ODE system is locally integrable in the neighborhood of all its stationary points, then it is globally integrable.

3.2. Local Integrability

Therefore, to find global integrability, we must first find all stationary points of the ODE system and then find out whether the system is locally integrable in their neighborhoods.

Let $X=(p,q,r,{\gamma }_1,{\gamma }_2,{\gamma }_3)$, the point $X=X^0$ is a stationary point in the system (1) and

$$M=\left(\begin{array}{cccccc}0& {\displaystyle \frac{B-C}{A}}r& {\displaystyle \frac{B-C}{A}}q& 0& -{\displaystyle \frac{z_0}{A}}& {\displaystyle \frac{y_0}{A}}\\ {\displaystyle \frac{C-A}{B}}r& 0& {\displaystyle \frac{C-A}{B}}p& {\displaystyle \frac{z_0}{B}}& 0& -{\displaystyle \frac{x_0}{B}}\\ {\displaystyle \frac{A-B}{C}}q& {\displaystyle \frac{A-B}{C}}p& 0& -{\displaystyle \frac{y_0}{C}}& {\displaystyle \frac{x_0}{C}}& 0\\ 0& -{\gamma }_3& {\gamma }_2& 0& r& -q\\ {\gamma }_3& 0& -{\gamma }_1& -r& 0& p\\ -{\gamma }_2& {\gamma }_1& 0& q& -p& 0\end{array}\right)$$

(18)

is a matrix of the linear part of the system (1) near the point $X^0$. The characteristic polynomial $\chi \left(\lambda \right)$ of the matrix M is $\chi \left(\lambda \right)={\lambda }^6+a_4{\lambda }^4+a_2{\lambda }^2$. Canceling it by ${\lambda }^2$ we obtain a bi-quadratic form, which primary discriminant on ${\lambda }^2$ is the following

$$D_{\lambda }\left(\chi \right)=a_4^2-4a_2.$$

(19)

It is a rational function $D=G/H$, where G and H are polynomials in system parameters.

A stationary point is locally integrable [9] if $a_2<0$ or $D_{\lambda }\left(\chi \right)<0$. But this property is not satisfied for definite values of system parameters (1).

The stationary points of the EP system form one-dimensional and two-dimensional families ${\mathcal{F}}_j^{\mathsf{\ell }}$ in ${\mathbb{R}}^6$. Below j is the number of the family for a given value of ℓ.

The numerator G of the first discriminant $D_{\lambda }\left(\chi \right)$ depends on the set $\Xi$ of parameters

$$\Xi \stackrel{\mathrm{def}}{=} \{s,A,B,C\}$$

(20)

where s is the parameter along the family ${\mathcal{F}}_j^{\mathsf{\ell }}$ and others are parameters of the system (1). Let $\xi$ be one of the parameters (20). By ${\Delta }_{\xi }\left({\mathcal{F}}_j^{\mathsf{\ell }}\right)$ we denote the secondary discriminant of the numerator G of the first discriminant (19) on the parameter $\xi$.

Hypothesis 2.

If near a stationary point $X^0$ of the family ${\mathcal{F}}_j^{\mathsf{\ell }}$ with certain parameters values (20) the EP equations are locally integrable, then at these parameter values there is at least one secondary discriminant ${\Delta }_{\xi }\left({\mathcal{F}}_j^{\mathsf{\ell }}\right)=0$.

Remark 1.

The property given in Hypothesis 2 is a necessary condition for local integrability, but not a sufficient one. So, the set of parameters for which Hypothesis 2 is satisfied should be called potentially integrable. Our first task is to compute all sets of potential integrability and then select from them the cases of local or global integrability.

We have found some general properties of the integrable cases 1–5, which were formulated as Hypothesis 2. So we have to compute all the values of the parameters $A,B,C$, $x_0,y_0,z_0$ for which this property and the triangle inequalities (2) are satisfied. Then, by computing the resonant terms of the normal form of the EP equations, extract from them those values at which the EP equations are integrable.

3.3. Zero Eigenvalues

The case where the characteristic polynomial $\chi \left(\lambda \right)$ has zero roots can be studied using the approach from [11]. But here we assume the following Hyposethis.

Hypothesis 3.

For EP equations the stationary point with four zero eigenvalues is locally integrable.

3.4. Checking for Integrability

There are three ways to check the existence of the fourth integral.

1.

Finding this integral in explicit form.

2.

Using the Normal Form (NF) of the system near the SP; In the last case, according to [Section 5.3] of [9] the coefficients of the resonance terms of the NF at a resonance of order 3, i.e., when there exists a pair of eigenvalues with ratio 2:1, should be zero in integrable cases. They are zero in some subcases, and in other subcases, they are nonzero. If resonance of order 3 is absent, then we need to consider the resonance with the minimum possible order.

As far as the characteristic polynomial of the linear system of EP equations is the following $\chi \left(\lambda \right)={\lambda }^6+a_4{\lambda }^4+a_2{\lambda }_2$, all eigenvalues can be reordered in such a way

$${\lambda }_1,-{\lambda }_1,{\lambda }_2,-{\lambda }_2,0,0.$$

Resonant Condition of existence of any resonance between eigenvalues ${\lambda }_{1,2}$: ${\lambda }_2/{\lambda }_1=\mathfrak{q}$, $\mathfrak{q}\in \mathbb{N}$ takes the form

$${Res}_{\mathfrak{q}:1}\left(\chi \right) \stackrel{\mathrm{def}}{=} {\mathfrak{q}}^2{a}_4^2-\left({\mathfrak{q}}^2+1\right)a_2=0.$$

(21)

For nonzero eigenvalues the resonant equation ${\lambda }_i=p{\lambda }_j+q{\lambda }_k$, $p,q\in {\mathbb{N}}_0$ has nontrivial solutions for a resonance of order 3 in 4 cases:

• For $i=1$: $p=q=1$, $j=2$, $k=3$.
• For $i=2$: $p=q=1$, $j=1$, $k=4$.
• For $i=3$: $p=2$, $q=0$, $j=1$.
• For $i=4$: $p=2$, $q=0$, $j=2$.

So, for a check of integrability, it is sufficient to provide one step of the normalization transformation $X=QY$, where Q is a matrix composed of the eigenvectors of the matrix M and reduces the linear system of EP equations in a diagonal form. After the reduction, it is only necessary to check the coefficients of the resonant monomials $Y_j^p{Y}_k^q$ of the ith equation for each set of $j,k,p,q$ integer values given above.

3.

If in the case that the EP equations (1) do not have any resonances, one has to consider the eigenvalues at the stationary point of a family ${\mathcal{F}}_j^{\mathsf{\ell }}$. If all of them do not belong to the straight line that crosses the origin of the complex plane, then in the real case, the normalization transformation is analytic, and the system (1) is locally integrable. See [12] for Hamiltonian systems and [9] for the EP equations because they can be written as a Hamiltonian system [6] with the same set of eigenvalues.

4. Algorithm of Searching for Integrable Cases

Taking into account Hypothesis 1, now the search for integrable cases consists of the following steps.

Step 1 

Fix the number ℓ of non-zero parameters $x_0,y_0,z_0$ and find all families ${\mathcal{F}}_j^{\mathsf{\ell }}$ of stationary points.

Step 2 

Compute the discriminants (19) $D_{\lambda }\left(\chi \right)$ on the families ${\mathcal{F}}_j^{\mathsf{\ell }}$.

Step 3 

In the family ${\mathcal{F}}_j^{\mathsf{\ell }}$, calculate all secondary discriminants ${\Delta }_{\xi }\left({\mathcal{F}}_j^{\mathsf{\ell }}\right)$ of the numerators G of the first discriminants $D_{\lambda }\left(\chi \right)$, where $\xi \in \Xi$ defined in (20), and their factors. All irreducible factors we divide into two groups:

Ist group 

are factors not depending on s:

$${\phi }_1,\dots ,{\phi }_m.$$

(22)

IInd group 

are factors depending on s:

$${\psi }_1,\dots ,{\psi }_n.$$

For each pair with ${\psi }_i$, ${\psi }_k$, $1⩽i<k⩽n$, we compute its resultant $R_s^{ik}(A,B,C)$ over the parameter s and factorize it into irreducible factors ${\eta }^{ik}(A,B,C)$. So, we obtain a set ${\mathcal{S}}_j$ of factors (22) and factors ${\eta }^{ik}(A,B,C)$ which are not dependent on s.

For their roots, all stationary points of family ${\mathcal{F}}_j^{\mathsf{\ell }}$ are potentially locally integrable.

Step 4 

For fixed value ℓ and for each set

$$f_1(A,B,C),f_2(A,B,C),f_3(A,B,C),$$

where $f_j\subset {\mathcal{S}}_j$, we compute their common real roots $A_0,B_0,C_0$, satisfying the triangle inequalities (2). Such roots are potentially globally integrable cases according to Hypothesizes 1 and 2.

Step 5 

Check the obtained parameter values for potential integrability by computing the normal forms of the EP system near stationary points or by finding the fourth independent integral. Usually, it is sufficient to compute coefficients of the resonant monomials of the normal form in the case of the resonance of order 3, i.e., when there exists at least one pair of nonzero eigenvalues with ration ${\lambda }_j:{\lambda }_k=2$.

Organization of Computations of Step 5

All the computations described in the beginning of this Section were implemented in the CAS Maple 2024 of Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

Two different situations had to be implemented separately.

1.

If, for a family of stationary points, it is possible to analytically describe the set of eigenvalues of the characteristic equation of the matrix M (18) as a function of the parameter s, then the NF of the Euler–Poisson equations (1) is computed analytically for the whole family.

2.

If it is not possible to do so, then, after preliminary simplification of the obtained expressions, the parameter variation interval is set, the set of points on this interval is determined, other parameters of the system are calculated, and then the numerical normalization procedure is performed. Such calculations are carried out using high-precision arithmetic with 30 decimal digits. At each step, intermediate checks of the significance of the results obtained are performed.

Remark 2.

In many cases during computations, very cumbersome expressions should appear. To reduce these expressions a bit, we would remove constant numerical coefficients of factorized expressions and the content of multivariable polynomials. In these cases, we use the symbol “≅” but not the symbol “=”.

5. Families of Stationary Points of the Euler–Poisson Equations

Here and below, to reduce the complexity and awkwardness of the formulas obtained, we consider $Mg=1$.

All stationary points of the system (1) are obtained as solutions to the following algebraic subsystems

$$\begin{array}{cc}\hfill (C-B)qr& =y_0{\gamma }_3-z_0{\gamma }_2,\hfill \\ \hfill (A-C)pr& =z_0{\gamma }_1-x_0{\gamma }_3,\hfill \\ \hfill (B-A)pq& =x_0{\gamma }_2-y_0{\gamma }_1,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill 0& =r{\gamma }_2-q{\gamma }_3,\hfill \\ \hfill 0& =p{\gamma }_3-r{\gamma }_1,\hfill \\ \hfill 0& =q{\gamma }_1-p{\gamma }_2.\hfill \end{array}$$

(24)

The solutions to the subsystem (24) are

$${\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3={\displaystyle \frac{r}{k}},$$

(25)

where k is a real number and ${\gamma }_j$, $j=1,2,3$, satisfy the third integral (5). Let us find all real solutions for the subsystem (23) for the values $\mathsf{\ell }=0,1,2,3$.

5.1. Families of SP for the Case $\mathsf{\ell }=0$

Here, the subsystem (23) becomes homogeneous, and its solutions form two groups of families. The first consists of three families of one parameter with zero pairs of the values $(p,q,r)$ without any limits on the principal momenta $A,B,C$:

${\mathcal{F}}_1^0:$

$\left\{p=s,q=r=0,{\gamma }_1=s/k=\pm 1,{\gamma }_2={\gamma }_3=0\right\}$;

${\mathcal{F}}_2^0:$

$\left\{q=s,p=r=0,{\gamma }_2=s/k=\pm 1,{\gamma }_1={\gamma }_3=0\right\}$;

${\mathcal{F}}_3^0:$

$\left\{r=s,p=q=0,{\gamma }_3=s/k=\pm 1,{\gamma }_1={\gamma }_2=0\right\}$;

• where s is a parameter for all families. Under the permutations (10) and (16) in Section 2 families ${\mathcal{F}}_2^0$ and ${\mathcal{F}}_3^0$ are reduced to the family ${\mathcal{F}}_1^0$.

The second consists of the other three two-parameter families with additional conditions on the principal momenta $A,B,C$:

${\mathcal{F}}_4^0:$

$\left\{p=0,{\gamma }_1=0,{\gamma }_2=q/k,{\gamma }_3=r/k,{\gamma }_2^2+{\gamma }_3^2=1,B=C\right\}$, $q,r$ are parameters;

${\mathcal{F}}_5^0:$

$\left\{q=0,{\gamma }_2=0,{\gamma }_1=p/k,{\gamma }_3=r/k,{\gamma }_1^2+{\gamma }_3^2=1,A=C\right\}$, $p,r$ are parameters;

${\mathcal{F}}_6^0:$

$\left\{r=0,{\gamma }_3=0,{\gamma }_1=p/k,{\gamma }_2=q/k,{\gamma }_1^2+{\gamma }_2^2=1,A=B\right\}$, $p,q$ are parameters.

• The proof is evident.

Under the permutations (10) and (16) in Section 2, families ${\mathcal{F}}_5^0$ and ${\mathcal{F}}_6^0$ are reduced to the family ${\mathcal{F}}_4^0$.

5.2. Families of SP for $l=1,2,3$

Theorem 1.

For $\mathsf{\ell }=1$ System (23) has four families of SP:

$$\begin{array}{cc}\hfill {\mathcal{F}}_1^1:& \left\{p=s,q=r=0,{\gamma }_1=p/k=\pm 1,{\gamma }_2={\gamma }_3=0\right\};\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\mathcal{F}}_2^1:& \left\{p={\displaystyle \frac{x_0}{k(C-A)}},q=0,r={\displaystyle \frac{s}{k}},{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2=0,{\gamma }_3={\displaystyle \frac{r}{k}},A\ne C,x_0\ne 0\right\};\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\mathcal{F}}_3^1:& \left\{p={\displaystyle \frac{x_0}{k(B-A)}},r=0,q={\displaystyle \frac{s}{k}},{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3=0,A\ne B,x_0\ne 0\right\};\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\mathcal{F}}_4^1:& \left\{p={\displaystyle \frac{x_0}{k(B-A)}},{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3={\displaystyle \frac{r}{k}},A\ne B=C,x_0\ne 0\right\},\hfill \end{array}$$

(29)

where s, q, r are parameters. Under the permutation (14) in Section 2, families ${\mathcal{F}}_3^1\leftrightarrow {\mathcal{F}}_2^1$. The families ${\mathcal{F}}_1^1$ and ${\mathcal{F}}_4^1$ are invariant under automorphism.

For $\mathsf{\ell }=2$ System (23) has two families of SP:

$$\begin{array}{cc}\hfill {\mathcal{F}}_1^2& :\left\{p={\displaystyle \frac{x_0}{k(C-A)}},q={\displaystyle \frac{y_0}{k(C-B)}},r=s,{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3={\displaystyle \frac{r}{k}},A\ne C\ne B\right\};\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\mathcal{F}}_2^2& :\left\{p=-{\displaystyle \frac{x_0}{k(A+s)}},q=-{\displaystyle \frac{y_0}{k(B+s)}},r=0,{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3=0\right\},\hfill \end{array}$$

(31)

where s is a parameter, $s\ne -A,-B$. Under the permutation (10) the families ${\mathcal{F}}_1^2$ and ${\mathcal{F}}_1^2$ are invariant.

For $\mathsf{\ell }=3$ System (23) has one family of SP:

$${\mathcal{F}}_1^3:\left\{p=-{\displaystyle \frac{x_0}{k(A+s)}},q=-{\displaystyle \frac{y_0}{k(B+s)}},r=-{\displaystyle \frac{z_0}{k(C+s)}},{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3={\displaystyle \frac{r}{k}}\right\},$$

(32)

where s is a parameter, $s\notin \{-A,-B,-C\}$. Under any permutation, the family ${\mathcal{F}}_1^3$ is invariant.

All variables ${\gamma }_j$, $j=1,2,3$, satisfy the partial integral (5).

Proof. 

We provide the proof separately for each value of ℓ.

1.

Case $\mathsf{\ell }=\mathbf{1}$. Taking into account (25) and the third integral (5) the system (23) becomes as

$$\begin{array}{cc}\hfill (C-B)qr& =0,\hfill \\ \hfill (A-C)pr& =-x_0{\gamma }_3=-x_0{\displaystyle \frac{r}{k}},\hfill \\ \hfill (B-A)pq& =x_0{\gamma }_2=x_0{\displaystyle \frac{q}{k}}.\hfill \end{array}$$

(33)

Now we consider sub-cases where all multipliers of the first equation are equal to zero.

(a)

Let $q=r=0$. Then the system (33) is trivially satisfied. This is the family ${\mathcal{F}}_1^1$ (26) and along it p is a parameter, ${\gamma }_1=p/k=\sigma =\pm 1$, ${\gamma }_2={\gamma }_3=0$.

(b)

Let $q=0$, $r\ne 0$, $A\ne C$. Then according to the second equation in (33) one obtains $(A-C)p=-x_0/k$, i.e., $p=-{\displaystyle \frac{x_0}{k(A-C)}}$. This is the family ${\mathcal{F}}_2^1$ (27) and along it r is a parameter, ${\gamma }_1=p/k$, ${\gamma }_2=0$, ${\gamma }_3=r/k$, ${\gamma }_1^2+{\gamma }_3^2=1$ and $A\ne C$.

(c)

Let $r=0$, $q\ne 0$, $A\ne B$. Then similar to the previous item one obtains $p={\displaystyle \frac{x_0}{k(B-A)}}$ and the family ${\mathcal{F}}_3^1$ (28) and along it q is a parameter, ${\gamma }_1=p/k$, ${\gamma }_2=q/k$, ${\gamma }_3=0$, ${\gamma }_1^2+{\gamma }_2^2=1$ and $B\ne A$.

(d)

Let $B=C$, $q\ne 0$, $r\ne 0$. Then the system (33) reduced to the equation $(B-A)p=x/k$ with solution $p={\displaystyle \frac{x_0}{k(B-A)}}$. This is the family ${\mathcal{F}}_4^1$ (29) and along it there are parameters $q,r$, ${\gamma }_1=p/k$, ${\gamma }_2=q/k$, ${\gamma }_3=r/k$, ${\gamma }_1^2+{\gamma }_2^2+{\gamma }_3^2=1$, and $A\ne B=C$.

2.

Case $\mathsf{\ell }=\mathbf{2}$. The system (23) takes the form

$$\begin{array}{cc}\hfill (C-B)qr& =y_0{\gamma }_3=y_0{\displaystyle \frac{r}{k}},\hfill \\ \hfill (A-C)pr& =-x_0{\gamma }_3=-x_0{\displaystyle \frac{r}{k}},\hfill \\ \hfill (B-A)pq& =x_0{\gamma }_2-y_0{\gamma }_1=x_0{\displaystyle \frac{q}{k}}-y_0{\displaystyle \frac{p}{k}}.\hfill \end{array}$$

(34)

(a)

Case $r\ne 0$. Then canceling by r the first and second equations of System (34) one obtains a system of two equations

$$\begin{array}{cccc}\hfill (C-B)q& =y_0/k,\hfill & \hfill (A-C)p& =-x_0/k,\hfill \end{array}$$

(35)

so far as the third equation of (35) becomes a consequence of them. From the system above, we can obtain

$$p=-{\displaystyle \frac{x_0}{k(A-C)}},q={\displaystyle \frac{y_0}{k(C-B)}}.$$

This is the family ${\mathcal{F}}_1^2$ (30) and along it r is a parameter, conditions (25) and (5) are satisfied, and $A\ne C\ne B$.

(b)

Case $r=0$. Then the system (34) reduced to the only equation $(B-A)pq=x_{0}q/k-y_{0}p/k$, canceling it by $pq$ we obtain an equation

$$B-A={\displaystyle \frac{x_0}{kp}}-{\displaystyle \frac{y_0}{kq}}.$$

Its solution is the following:

$$B+s=-{\displaystyle \frac{y_0}{kq}},A+s=-{\displaystyle \frac{x_0}{kp}},$$

where s is a parameter, i.e.,

$$p=-{\displaystyle \frac{x_0}{k(A+s)}},q=-{\displaystyle \frac{y_0}{k(B+s)}}.$$

This is the family ${\mathcal{F}}_1^2$ (31) and with parameter s, and along it conditions (25) and (5) are satisfied.

3.

Case $\mathsf{\ell }=\mathbf{3}$. Here, we should solve the whole system (23). For this, let us cancel its first equation by $qr$, the second one by $pr$, and the third one by $pq$. So, we obtain a linear system

$$\begin{array}{cc}\hfill C-B& ={\displaystyle \frac{y_0}{kq}}-{\displaystyle \frac{z_0}{kr}},\hfill \\ \hfill A-C& ={\displaystyle \frac{z_0}{kr}}-{\displaystyle \frac{x_0}{kp}},\hfill \\ \hfill B-A& ={\displaystyle \frac{x_0}{kp}}-{\displaystyle \frac{y_0}{kq}}\hfill \end{array}$$

of the fractions $x_0/\left(kp\right)$, $y_0/\left(kq\right)$ and $z_0/\left(kr\right)$. The ranks of its coefficient and the augmented matrices are equal to 2. So its solutions are the following:

$${\displaystyle \frac{x_0}{kp}}=-(A+s),{\displaystyle \frac{y_0}{kq}}=-(B+s),{\displaystyle \frac{z_0}{kr}}=-(C+s),$$

where s is an arbitrary real value (parameter). So,

$$p=-{\displaystyle \frac{x_0}{k(A+s)}},q=-{\displaystyle \frac{y_0}{k(B+s)}},r=-{\displaystyle \frac{z_0}{k(C+s)}}.$$

This is the family ${\mathcal{F}}_1^3$ (32), it is unique, and identities (25) and (5) take place on it.

□

Finally, the EP equations have 13 families of SP, and they should be checked for local integrability.

5.3. Permutation Symmetries and Families of SP

Let us apply the permutation group of EP equations, described in the introduction, to reduce the number of families of SP that should be checked for local integrability.

For $\mathsf{\ell }=0$ any elementary permutation of a pair of columns in the matrix $N_1$ transform the family ${\mathcal{F}}_1^0$ into one of the other families ${\mathcal{F}}_j^0$, $j=2,3$. The same happens for the families ${\mathcal{F}}_j^0$, $j=4,5,6$.

For $\mathsf{\ell }=1$ the families ${\mathcal{F}}_2^1$ and ${\mathcal{F}}_3^1$ are the same up to exchange $B\leftrightarrow C$ and $q\leftrightarrow r$.

So, we should study the local integrability only of 8 families ${\mathcal{F}}_1^0$, ${\mathcal{F}}_4^0$, ${\mathcal{F}}_1^1$, ${\mathcal{F}}_2^1$, ${\mathcal{F}}_4^1$, ${\mathcal{F}}_1^2$, ${\mathcal{F}}_2^2$, and ${\mathcal{F}}_1^3$ from the entire list of 13 families.

6. Results for Case $\mathsf{\ell }=\mathbf{0}$

This is the Case 1 of Euler–Poinsot with additional integral (6). So, here we only demonstrate the correctness of hypotheses 2 and 3.

For family ${\mathcal{F}}_1^0$ we have the characteristic polynomial $\chi \left(\lambda \right)$ as follows

$$\chi \left({\mathcal{F}}_1^0\right)={\lambda }^6+{\displaystyle \frac{s^2\left(A^2-AB-AC+2BC\right)}{CB}}{\lambda }^4+{\displaystyle \frac{s^4\left(A-B\right)\left(A-C\right)}{CB}}{\lambda }^2,$$

(36)

numerator of its discriminant over $\lambda$ is

$$G\left({\mathcal{F}}_1^0\right)=A^2{(A-B-C)}^2{s}^4.$$

It means that any of the secondary discriminant ${\Delta }_{\xi }\left({\mathcal{F}}_1^0\right)=0$ for $\xi \in {\Xi }_0$. The coefficient $a_2$ of the polynomial (36) degenerates at $A=C$ or $B=C$. In this case there are four zero eigenvalues, but the matrix $M\left({\mathcal{F}}_1^0\right)$ has only one Jordan block $2\times 2$, and two other zero eigenvalues are semi-simple. It is evident that the conditions of Hypothesis 2 and 3 are satisfied.

For family ${\mathcal{F}}_4^0$, the characteristic polynomial $\xi \left(\lambda \right)$ becomes degenerate:

$$\chi \left({\mathcal{F}}_4^0\right)={\lambda }^6+\left(q^2+r^2\right){\lambda }^4,$$

and has four zero eigenvalues. The matrix $M\left({\mathcal{F}}_4^0\right)$ has only one $2\times 2$ Jordan block, and two other zero eigenvalues are semi-simple.

Here we have the fourth integral (6), and the NF computation step should be skipped.

7. Results for Case $\mathsf{\ell }=\mathbf{1}$

Let us apply this approach to the case $\mathsf{\ell }=1$. In this case $x_0\ne 0,y_0=z_0=0$. Now we study local integrability for families ${\mathcal{F}}_1^1$, ${\mathcal{F}}_2^1$, ${\mathcal{F}}_3^1$ and ${\mathcal{F}}_4^1$.

7.1. Conditions of Potential Integrability for Family ${\mathcal{F}}_1^1$

Starting from the “simplest” family ${\mathcal{F}}_1^1$ and obtaining a set of relations $\mathcal{S}\left({\mathcal{F}}_1^1\right)$ between parameters ${\Xi }_1$, we would provide the corresponding computations for other families only taking into account the relations obtained for the families processed earlier.

So, for the family ${\mathcal{F}}_1^1$ we have the following:

• coefficients $a_4$ and $a_2$ of the characteristic polynomial $\chi \left({\mathcal{F}}_1^1\right)$ are

  $$a_4={\displaystyle \frac{\left(B+C\right)x_0+s^2\left(A^2-AB-AC+2CB\right)}{BC}},a_2={\displaystyle \frac{\left((A-C)s^2+x_0\right)\left((A-B)s^2+x_0\right)}{BC}};$$

  (37)
• numerator G of the discriminant $D_{\lambda }\left(\chi \right)$ is

  $$G\left({\mathcal{F}}_1^1\right)=A^2{\left(A-C-B\right)}^2{s}^4+2\left(BA+AC-4CB\right)\left(A-C-B\right)x_0{s}^2+{\left(B-C\right)}^2{x}_0^2,$$

  (38)
• the secondary discriminants ${\Delta }_{\xi }$ are the following

  $$\begin{array}{cc}\hfill {\Delta }_{s^2}\left({\mathcal{F}}_1^1\right)& \cong {\left(A-2C\right)}^2{\left(A-2B\right)}^2{\left(A-B-C\right)}^6{\left(B-C\right)}^2{A}^2{B}^2{C}^2{x}_0^6,\hfill \\ \hfill {\Delta }_A\left({\mathcal{F}}_1^1\right)& \cong g_{1A}{\left(B-C\right)}^2{x}_0^3{B}^2{C}^2{s}^{12},\hfill \\ \hfill {\Delta }_B\left({\mathcal{F}}_1^1\right)& \cong {\left(A-2C\right)}^2{g}_{1B}{x}_0^{2}C{s}^2.\hfill \end{array}$$
• According to permutation $B\leftrightarrow C$ we obtain

  $${\Delta }_C\left({\mathcal{F}}_1^1\right)\cong {\left(A-2B\right)}^2{g}_{1C}{x}_0^{2}B{s}^2,$$

  where

  $$\begin{array}{cc}\hfill g_{1A}=& 2{\left(B+C\right)}^3{s}^6+3\left(C-5B\right)\left(B-5C\right)s^4{x}_0+\left(24B+24C\right)s^2{x}_0^2+16x_0^3,\hfill \\ \hfill g_{1B}=& (A-C)s^2+x_0,g_{1C}=(A-B)s^2+x_0.\hfill \end{array}$$

7.2. Conditions of Potential Integrability for Families ${\mathcal{F}}_2^1$, ${\mathcal{F}}_3^1$

For the family ${\mathcal{F}}_2^1$ we have the following.

The coefficients of the characteristic polynomial $\chi \left({\mathcal{F}}_2^1\right)$ are

$$\begin{array}{cc}\hfill a_4& ={\displaystyle \frac{\left(2AB-(A+B)C+C^2\right)s^2}{AB{k}^2}}+{\displaystyle \frac{\left({(A-C)}^2+(3C-2A)B\right)x_0^2}{{\left(A-C\right)}^{2}BC{k}^2}},\hfill \\ \hfill a_2& ={\displaystyle \frac{\left(C{(A-C)}^2{s}^2+(4C-3A)x_0^2\right)\left(B-C\right)s^2}{ABC\left(A-C\right)k^4}}.\hfill \end{array}$$

The connection between the parameter k and other parameters of the set $\Xi$ is

$${\left(A-C\right)}^2{k}^4=x_0^2+s^2{\left(A-C\right)}^2.$$

The numerator G of the discriminant $D_{\lambda }\left({\mathcal{F}}_2^1\right)$ is the following.

$$\begin{array}{c}G\left({\mathcal{F}}_2^1\right)=C^4{\left(A-C\right)}^4{\left(A+B-C\right)}^2{s}^4+2AC{\left(A-C\right)}^2\left(A+B-C\right)\left(2A^{2}B-A^{2}C-6ABC+2A{C}^2+5B{C}^2-C^3\right)s^2{x}_0^2\hfill \\ \hfill +A^2{\left({(A-C)}^2+(3C-2A)B\right)}^2{x}_0^4 \end{array}$$

and the secondary discriminants ${\Delta }_{\xi }$ are

$$\begin{array}{cc}\hfill {\Delta }_{s^2}\left({\mathcal{F}}_2^1\right)\cong & {\left(B-C\right)}^2{\left(A-2C\right)}^4{\left(A-C\right)}^{16}{\left(A+B-C\right)}^6{\left({(A-C)}^2+(3C-2A)B\right)}^2{A}^6{B}^2{C}^8{x}_0^{12},\hfill \\ \hfill {\Delta }_A\left({\mathcal{F}}_2^1\right)\cong & {(B-C)}^5{h}_{2A}{g}_{2A}^2{B}^8{C}^{19}{x}_0^{20}{s}^8,\hfill \\ \hfill {\Delta }_B\left({\mathcal{F}}_2^1\right)\cong & {\left(A-2C\right)}^2{\left(A-C\right)}^5{g}_{2B}{A}^{3}C{x}_0^4{s}^2,\hfill \\ \hfill {\Delta }_C\left({\mathcal{F}}_2^1\right)\cong & f_{13}(s,x_0,A,B)g_{2C}^2{A}^{25}{B}^{10}{x}_0^{28}{s}^{26},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill h_{2A}=& 16B^5{C}^7{s}^8-8C^5{B}^3\left(5B^2-10BC-27C^2\right)s^6{x}_0^2+3B{C}^3\left(291B^4+4C{B}^3-638B^2{C}^2+612B{C}^3+243C^4\right)s^4{x}_0^4\hfill \\ \hfill & -8BC\left(3B+C\right)\left(9B^3+249B^{2}C-557B{C}^2+171C^3\right)s^2{x}^6+16{\left(3B+C\right)}^4{x}_0^8,\hfill \\ \hfill g_{2A}=& 2(B-C)x_0^2+C^2(B+C)s^2,g_{2B}=\left(4C-3A\right)x_0^2+C{\left(A-C\right)}^2{s}^2,g_{2C}=8\left(2B-A\right)x_0^2+A^2\left(A+2B\right)s^2,\hfill \end{array}$$

and $f_{13}(s,x_0,A,B)$ can be presented in the following form:

$$\begin{array}{c}\hfill f_{13}=432B^7{A}^6{\left(A-B\right)}^2{\left(A+B\right)}^6{s}^{14}+8b_{12}{A}^4{B}^5{\left(A+B\right)}^2{s}^{12}{x}_0^2+b_{10}{A}^2{B}^3{s}^{10}{x}_0^4+b_{8}B{s}^8{x}_0^6-\\ \hfill b_{6}B{s}^6{x}_0^8+b_{4}A{s}^4{x}_0^{10}-b_2{A}^2{s}^2{x}_0^{12}+729A^3\left(A+4B\right)\left(A-2B\right){\left(4A-9B\right)}^2{x}_0^{14},\end{array}$$

(39)

where

$$\begin{array}{cc}\hfill b_{12}=& 761A^8+1688B{A}^7-1618A^6{B}^2-3970B^3{A}^5-3190B^4{A}^4+8468B^5{A}^3+1198A^2{B}^6-1658A{B}^7-815B^8,\hfill \\ \hfill b_{10}=& 23139A^{12}+132176B{A}^{11}+66014A^{10}{B}^2-462912A^9{B}^3-903427B^4{A}^8+66216B^5{A}^7+2109404A^6{B}^6+\hfill \\ \hfill & 541256B^7{A}^5-1110043B^8{A}^4-585752B^9{A}^3+131910B^{10}{A}^2+24984B^{11}A-11205B^{12},\hfill \\ \hfill b_8=& 11664A^{14}+38124B{A}^{13}-515985B^2{A}^{12}-2100641B^3{A}^{11}-1459987A^{10}{B}^4+488055A^9{B}^5+14807952B^6{A}^8+\hfill \\ \hfill & 1520778B^7{A}^7-21198114B^8{A}^6+1901814B^9{A}^5+6122283B^{10}{A}^4-2751305B^{11}{A}^3-878715B^{12}{A}^2+\hfill \\ \hfill & 675783B^{13}A+354294B^{14},\hfill \\ \hfill b_6=& 194904A^{12}+375197B{A}^{11}-3845225A^{10}{B}^2-5587486A^9{B}^3-14957233B^4{A}^8-31548982B^5{A}^7+\hfill \\ \hfill & 188909855A^6{B}^6-121115050B^7{A}^5-47215663B^8{A}^4+50281061B^9{A}^3+27641898B^{10}{A}^2-32393844B^{11}A-1417176B^{12},\hfill \\ \hfill b_4=& 256A^{10}+1030033A^{9}B-1258031A^8{B}^2-5051707A^7{B}^3+57371906A^6{B}^4-217988677A^5{B}^5-\hfill \\ \hfill & 45552939A^4{B}^6+1106080247A^3{B}^7-1327829952A^2{B}^8+440219448A{B}^9+111676968B^{10},\hfill \\ \hfill b_2=& 3456A^7+1644948A^{6}B-24256413B^2{A}^5+72436036B^3{A}^4+211335983B^4{A}^3-1492372490A^2{B}^5+\hfill \\ \hfill & 2740611492A{B}^6-1690806312B^7.\hfill \end{array}$$

According to permutation $B\leftrightarrow C$ all secondary discriminants ${\Delta }_{\xi }$ for family ${\mathcal{F}}_3^1$ are symmetric to the corresponding discriminants of the family ${\mathcal{F}}_2^1$ (see (42) and (43) below).

7.3. Local Integrability for the Family ${\mathcal{F}}_4^1$

For the family ${\mathcal{F}}_4^1$ of SP, the coefficient $a_2$ of the characteristic polynomial $\chi \left({\mathcal{F}}_4^1\right)$ is equal to zero. So, we have four zero eigenvalues here. The matrix $M\left({\mathcal{F}}_4^1\right)$ has only one Jordan block $2\times 2$, and Hypothesis 3 is applicable and gives local integrability for the family ${\mathcal{F}}_4^1$.

7.4. Potentially Integrable Cases for $\mathsf{\ell }=1$

According to Section 7.1 and Section 7.2 local integrabilities for families ${\mathcal{F}}_j^1$, $j=1,2,3$, are

$$\begin{array}{cc}\hfill {\mathcal{F}}_1^1:& L_{11}=\{B=C\},L_{12}=\{A=2C\},L_{13}=\{A=2B\},L_{14}=\{A=B+C\};\hfill \\ \hfill {\mathcal{F}}_2^1:& L_{21}=\{B=C\};L_{22}=\{A=2C\},L_{23}=\{A=C\},L_{24}=\{C=A+B\};\hfill \\ \hfill & L_{25}=\left\{A^2-2AB-2AC+3BC+C^2=0\right\}.\hfill \\ \hfill {\mathcal{F}}_3^1:& L_{31}=\{B=C\},L_{32}=\{A=2B\},L_{33}=\{A=B\},L_{34}=\{B=A+C\},\hfill \\ \hfill & L_{35}=\left\{A^2-2AB-2AC+3BC+B^2=0\right\},\hfill \end{array}$$

(40)

For all families ${\mathcal{F}}_j^1$, $j=1,2,3$, there is a case $L_{11}\equiv L_{21}\equiv L_{31}$, i.e., $B=C$. So, it is an integrable case. For families ${\mathcal{F}}_1^1$ and ${\mathcal{F}}_2^1$, there are cases $L_{12}\equiv L_{22}$, i.e., $A=2C$, and for family ${\mathcal{F}}_3^1$ there is a case $L_{33}$, i.e., $A=B$. So, case $A=B=2C$ is integrable. Similarly, case $A=C=2B$ is also integrable. These cases of integrability are known.

In addition to these cases, there is one more case $L_{13}$, $L_{24}$, $L_{32}$, that is, $A=2B$, $C=3B$ or

$$A=2B,B=C/3.$$

(41)

Any other combinations of local integrability cases contradict the triangle inequalities (2), or give the case $B=C$.

7.5. Some New Potentially Integrable Cases

In Section 7.4 we considered only factors that do not depend on s. Let us take into account factors which do depend on the parameters mentioned above:

$$\begin{array}{cc}\hfill {\mathcal{F}}_1^1& :g_{1A},g_{1B},g_{1C};\hfill \\ \hfill {\mathcal{F}}_2^1& :g_{2A},g_{2B},g_{2C},h_{2A},f_{13}(s,x_0,A,B);\hfill \\ \hfill {\mathcal{F}}_3^1& :g_{3A},g_{3B},g_{3C},h_{3A},f_{13}(s,x_0,A,C),\hfill \end{array}$$

(42)

where

$$g_{3A}(B,C)=g_{2A}(C,B),g_{3B}(B,C)=g_{2C}(C,B),g_{3C}(B,C)=g_{2B}(C,B),h_{3A}(B,C)=h_{2A}(C,B).$$

(43)

Here we want to note that the parameter s along one family is independent of the parameters along the other one. So, it can be eliminated from the pairs of factors related to the same family of SP only.

Now we provide a brief description of the computational procedure, which consists of the following steps.

Step 1 

Using the elimination technique, we prepare sets of polynomial factors for each family ${\mathcal{F}}_j^1$, $j=1,2,3$, eliminating parameters s from the polynomials in Formula (42).

Step 2 

The sets ${\mathcal{S}}_j$ of possible factors are prepared by union polynomials from Formula (40) with factors obtained in the previous step.

Step 3 

From each factor of the corresponding set ${\mathcal{S}}_j$, $j=1,2,3$, we construct a system of polynomial equations, compute the Gröbner basis for it and find all real non-trivial solutions, that is, all parameters $A,B,C$ are non-zero.

Step 4 

The solutions obtained are checked for whether they satisfy the triangle inequalities (2).

To implement Step 1 for family ${\mathcal{F}}_1^1$, one just needs to calculate the elimination ideals for pairs of ideals $⟨g_{1A},g_{1B}⟩$ and $⟨g_{1A},g_{1C}⟩$ and choose only those factors that depend on the parameters $A,B,C$ only. So, here we obtain two polynomials

$$\begin{array}{cc}\hfill h_{11}=& 16A^3-24A^{2}B-72A^{2}C-15A{B}^2+126ABC+81C^{2}A-2B^3+9B^{2}C-108B{C}^2-27C^3,\hfill \\ \hfill h_{12}=& 16A^3-72A^{2}B-24A^{2}C+81A{B}^2+126ABC-15C^{2}A-27B^3-108B^{2}C+9B{C}^2-2C^3.\hfill \end{array}$$

Eliminating s from the pair $g_{1B}$ and $g_{1C}$ gives only the $B-C$ value found above.

For the family ${\mathcal{F}}_2^1$ using six pairs of polynomials from the set $\{g_{2A},g_{2B},g_{2C},h_{2A}\}$, we obtain six new polynomials, some of them are factorized:

$$\begin{array}{cc}\hfill h_{21}=& 2A^{2}B-2A^{2}C-ABC+7C^{2}A-2B{C}^2-6C^3,\hfill \\ \hfill h_{22}=& \left(A-2C\right)\left(A^{2}B-A^{2}C+2A{B}^2-2C^{2}A+4B^{2}C+4B{C}^2\right),\hfill \\ \hfill h_{23}=& 3A^4+6A^{3}B-12A^{3}C+8A^{2}BC+16A^2{C}^2-32AB{C}^2-8A{C}^3+16B{C}^3,\hfill \\ \hfill h_{24}=& \left(5B^3-10B^{2}C+5B{C}^2+4C^3\right)\left(3B+C\right)\left(5B^2+10BC+C^2\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill h_{25}=& 1296A^8{B}^4+1728A^8{B}^{3}C+864A^8{B}^2{C}^2+192A^{8}B{C}^3+16A^8{C}^4-648A^7{B}^5-28512A^7{B}^{4}C+\hfill \\ \hfill & 20304A^7{B}^3{C}^2-5856A^7{B}^2{C}^3-5640A^{7}B{C}^4-128A^7{C}^5+12609A^6{B}^{5}C+169452A^6{B}^4{C}^2-\hfill \\ \hfill & 219114A^6{B}^3{C}^3+32972A^6{B}^2{C}^4+42033A^{6}B{C}^5+448A^6{C}^6-68364A^5{B}^5{C}^2-488448A^5{B}^4{C}^3+\hfill \\ \hfill & 808848A^5{B}^3{C}^4-134256A^5{B}^2{C}^5-148884A^{5}B{C}^6-896A^5{C}^7+178614A^4{B}^5{C}^3+805512A^4{B}^4{C}^4-\hfill \\ \hfill & 1596876A^4{B}^3{C}^5+323016A^4{B}^2{C}^6+298614A^{4}B{C}^7+1120A^4{C}^8-262404A^3{B}^5{C}^4-800544A^3{B}^4{C}^5+\hfill \\ \hfill & 1875840A^3{B}^3{C}^6-452368A^3{B}^2{C}^7-359628A^{3}B{C}^8-896A^3{C}^9+224545A^2{B}^5{C}^5+473644A^2{B}^4{C}^6-\hfill \\ \hfill & 1323018A^2{B}^3{C}^7+365772A^2{B}^2{C}^8+258609A^{2}B{C}^9+448A^2{C}^{10}-105824A{B}^5{C}^6-152960A{B}^4{C}^7+\hfill \\ \hfill & 520512A{B}^3{C}^8-158976A{B}^2{C}^9-102624AB{C}^{10}-128A{C}^{11}+21488B^5{C}^7+20560B^4{C}^8-88224B^3{C}^9+\hfill \\ \hfill & 28832B^2{C}^{10}+17328B{C}^{11}+16C^{12},\hfill \\ \hfill h_{26}=& 81A^{12}{B}^4+108A^{12}{B}^{3}C+54A^{12}{B}^2{C}^2+12A^{12}B{C}^3+A^{12}{C}^4+648A^{11}{B}^5+864A^{11}{B}^{4}C+432A^{11}{B}^3{C}^2+\hfill \\ \hfill & 96A^{11}{B}^2{C}^3+8A^{11}B{C}^4+1944A^{10}{B}^6+2484A^{10}{B}^{5}C-1728A^{10}{B}^4{C}^2+5976A^{10}{B}^3{C}^3+200A^{10}{B}^2{C}^4-\hfill \\ \hfill & 684A^{10}B{C}^5+2592A^9{B}^7+3024A^9{B}^{6}C-10368A^9{B}^5{C}^2+23136A^9{B}^4{C}^3+736A^9{B}^3{C}^4-2736A^9{B}^2{C}^5+\hfill \\ \hfill & 1296A^8{B}^8+1728A^8{B}^{7}C+864A^8{B}^6{C}^2+3684A^8{B}^5{C}^3+64A^8{B}^4{C}^4-7656A^8{B}^3{C}^5+7344A^8{B}^2{C}^6+\hfill \\ \hfill & 2916A^{8}B{C}^7+1728A^7{B}^{8}C+48384A^7{B}^7{C}^2-91008A^7{B}^6{C}^3-2816A^7{B}^5{C}^4+10944A^7{B}^4{C}^5+\hfill \\ \hfill & 1728A^6{B}^{9}C+48384A^6{B}^8{C}^2-118944A^6{B}^7{C}^3-3200A^6{B}^6{C}^4+70912A^6{B}^5{C}^5-56192A^6{B}^4{C}^6-\hfill \\ \hfill & 16416A^6{B}^3{C}^7+5120A^5{B}^6{C}^5-10240A^5{B}^5{C}^6-27648A^5{B}^4{C}^7+55872A^4{B}^9{C}^3+768A^4{B}^8{C}^4-\hfill \\ \hfill & 122496A^4{B}^7{C}^5+117504A^4{B}^6{C}^6+50752A^4{B}^5{C}^7-20480A^3{B}^8{C}^5+40960A^3{B}^7{C}^6+77824A^3{B}^6{C}^7+\hfill \\ \hfill & 20480A^2{B}^9{C}^5-40960A^2{B}^8{C}^6-12288A^2{B}^7{C}^7-131072A{B}^8{C}^7+65536B^9{C}^7.\hfill \end{array}$$

For family ${\mathcal{F}}_3^1$ similarly using six pairs of polynomials from the set $\{g_{3A},g_{3B},g_{3C},h_{3A}\}$ we obtain six new polynomials. All of them can be obtained from the above polynomials with symmetry transformation $B\leftrightarrow C$.

In Step 2 we prepare 3 sets ${\mathcal{S}}_j$, $j=1,2,3$, which contain all possible factors for each family, namely 6 factors in ${\mathcal{S}}_1$ (4 factors from (40) and 2 factors found above) and 14 factors in each ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ set (5 factors from (40) and 9 factors found above). In total, we obtain $6·14·14=1176$ possible combinations of factors.

The implementation of Steps 3 and 4 gave us 17 non-trivial solutions, i.e., solutions with non-zero values of $A,B,C$ satisfying the triangle inequalities (2). Nine of these solutions correspond to Case 2 of Lagrange–Poisson: one common case and eight particular cases. Two solutions correspond to Case 3 of Kovalevskaya up to the permutation $B\leftrightarrow C$, $q\leftrightarrow r$. Two solutions correspond to the case (41) considered in Section 7.4. So, we could find two new pairs of solutions:

1.

$A=2C$, $B=3C/2$ and symmetrical

$$A=2B,B=2C/3.$$

(44)

2.

$A=2C$, $3B^3-6B^{2}C+5B{C}^2-4C^3=0$ and symmetrical under permutation $B\leftrightarrow C$ is $A=2B$, $4B^3-5B^{2}C+6B{C}^2-3C^3=0$. The last homogeneous cubic equation has only one real root for

$$B/C={\displaystyle \frac{\alpha -47/\alpha +5}{12}}\approx 0.67456267248393436447,$$

(45)

where $\alpha =\sqrt[3]{233++36\sqrt{122}}$.

Up to this point, we did not use the polynomial $f_{13}$ (39) as a source of new potentially integrable cases due to its complexity. Now it is time to involve them in computations, but in reduced form. From (40) it is evident that the families ${\mathcal{F}}_1^1$ and ${\mathcal{F}}_3^1$ have the only common condition $A=2B$. Let us apply this condition to the polynomials $g_{2B}$, $g_{2C}$ and $f_{13}$ depending on the parameter A, as soon as the polynomials $h_{2A}$ and $g_{2A}$ do not depend on it. After that we obtain such reduced versions of these polynomials, further denoted as ${\tilde{g}}_{2B}$, ${\tilde{g}}_{2C}$, and ${\tilde{f}}_{13}$ correspondingly. The polynomial ${\tilde{g}}_{2C}$ reduces to $16B^2{s}^2$, which is not applicable; ${\tilde{f}}_{13}$ is factorized and after canceling by the trivial term, we obtain the following.

$$\begin{array}{cc}\hfill {\tilde{g}}_{2B}=& 2\left(2C-3B\right)x^2+C{\left(2B-C\right)}^2{s}^2,\hfill \\ \hfill {\tilde{f}}_{13}=& 559872B^{12}{s}^{12}+6324896B^{10}{s}^{10}{x}^2+11522363B^8{s}^8{x}^4-122041776B^6{s}^6{x}^6+144255128B^4{s}^4{x}^8\hfill \\ \hfill & +3095552B^2{s}^2{x}^{10}-11664x^{12}\hfill \end{array}$$

Now we eliminate the parameter s from each pair $\left(\zeta ,{\tilde{f}}_{13}\right)$, where $\zeta \in \{h_{2A},g_{2A},{\tilde{g}}_{2B}\}$. After this we obtain three new quasi-homogeneous polynomials of $B,C$, and then for each such polynomial, we compute real positive roots. The first pair eliminator is a polynomial of degree 42, but all its real roots are negative, and we omit it. So, here we present the polynomial $h_{27}$ as a result of elimination ${\tilde{f}}_{13}$ and $g_{2A}$:

$$\begin{array}{c}\hfill h_{27}=2239488B^{18}-13436928B^{17}C+20942528B^{16}{C}^2+5809408B^{15}{C}^3-18134277B^{14}{C}^4-36481654B^{13}{C}^5\\ \hfill +114986973B^{12}{C}^6-4509716B^{11}{C}^7-145871453B^{10}{C}^8+49082838B^9{C}^9+158134301B^8{C}^{10}-145802904B^7{C}^{11}\\ \hfill -99020119B^6{C}^{12}+72123190B^5{C}^{13}+37987567B^4{C}^{14}+1533196B^3{C}^{15}+376009B^2{C}^{16}-4374B{C}^{17}-729C^{18},\end{array}$$

and the polynomial $h_{28}$ as a result of elimination ${\tilde{f}}_{13}$ and $g_{2B}$:

$$\begin{array}{c}\hfill h_{28}=1632586752B^{18}+5765250816B^{17}C-27464029776B^{16}{C}^2-86099607072B^{15}{C}^3+655687219464B^{14}{C}^4\\ \hfill -1714227420312B^{13}{C}^5+2658633663915B^{12}{C}^6-2784730679784B^{11}{C}^7+2072225505960B^{10}{C}^8\\ \hfill -1114722935592B^9{C}^9+431073080416B^8{C}^{10}-116129030720B^7{C}^{11}+20135789078B^6{C}^{12}-1770489416B^5{C}^{13}\\ \hfill -24035032B^4{C}^{14}+17921632B^3{C}^{15}-966344B^2{C}^{16}+17496B{C}^{17}-729C^{18}.\end{array}$$

Among the set of roots obtained, we need to select only those that satisfy the triangle inequalities (2). The polynomials $h_{27}$ and $h_{28}$ have four real roots each, among them, exactly three are positive. The triangular inequalities (2) are successful on the following values $B/C$ only:

$$B/C\in \{0.66566380248467780658,0.66680311817903079098,0.99671298306063041299\}.$$

Summarizing the results of potentially integrable cases obtained during the computations in Section 7.4 and Section 3.4, we can state that all the cases are implemented under the conditions $A=2B$ and $B/C=\beta$, where the parameter $\beta$ takes its values from the set

$$\mathcal{B} \stackrel{\mathrm{def}}{=} \{1/3,0.6656638024846778,2/3,0.6668031181790308,0.6745626724839344,0.9967129830606304\}.$$

(46)

For each of these values, we need to provide the check of local integrability separately for the families ${\mathcal{F}}_1^1$ and ${\mathcal{F}}_2^1$. Here we meet three different situations concerning the Resonant condition (21) for the definite value of $\beta$.

1.

Resonance of order 3 is implemented for some real values of the parameter $x_0$.

2.

Resonance of order 3 does not exist for any real values of the parameter $x_0$, but it takes place for some resonances of higher order.

3.

There are no resonances of any orders for any real values of the parameter $x_0$.

In the first situation, we only need to provide the NF computation as described in Section 3.4.

In the second situation it is necessary to compute the NF of the EP equations up to the minimal possible order and then check the coefficients of the resonant monomials. In symbolical form such computations look more or less possible to provide only for resonances of very small orders, say 4 or 5, because of the fact that the complexity of the normalizing procedure grows exponentially while the order of the resonance increases.

Finally, Situation 3 changes the strategy of computations. Here we just need to prove, according to Moser’s Theorem [12], that all eigenvalues of the linear part of the EP equations do not position on the same straight line crossing the origin of the complex plane.

In the following section, Section 7.6, we demonstrate in detail the NF computation for the case $A=2B$ and $B/C=\beta$ whenever it is possible to compute it symbolically.

7.6. Normal Form Computations for the Case $A=2B$, $B/C=\beta$

In the case being considered, the triangle inequalities (2) imply that $1/3⩽\beta ⩽1$.

For the family ${\mathcal{F}}_1^1$ we obtain the following:

•

the matrix M of the linear system in (18) becomes

$$M\left({\mathcal{F}}_1^1\right)=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& -{\displaystyle \frac{\left(2\beta -1\right)s}{\beta }}& 0& 0& -{\displaystyle \frac{x_0}{\beta C}}\\ 0& \beta s& 0& 0& {\displaystyle \frac{x_0}{C}}& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& \mp 1& 0& 0& s\\ 0& \pm 1& 0& 0& -s& 0\end{array}\right),$$

•

coefficients of the characteristic polynomial (37) are reduced into

$$a_4=2\beta s^2+{\displaystyle \frac{\left(\beta +1\right)x_0}{C\beta }},a_2={\displaystyle \frac{\left((2\beta -1)C{s}^2+x_0\right)\left(C\beta s^2+x_0\right)}{\beta C^2}},$$

•

Resonant Condition (21) for the resonance $2:1$ existence takes the form

$${Res}_{2:1}\left({\mathcal{F}}_1^1\right) \stackrel{\mathrm{def}}{=} C^2\left((2\beta -5)C\beta s^2+(\beta -4)x_0\right)\left((8\beta -5)C\beta s^2+(4\beta -1)x_0\right)$$

which immediately gives two families of solutions:

$$x_0^{\left(1\right)}={\displaystyle \frac{C\beta s^2\left(2\beta -5\right)}{4-\beta }},x_0^{\left(2\right)}={\displaystyle \frac{C\beta s^2\left(8\beta -5\right)}{1-4\beta }}.$$

For the first family $x_0^{\left(1\right)}$ the set of eigenvalues is

$$\mathit{\lambda }=\{\pm i2{\varkappa }_{1}s,\pm i{\varkappa }_{1}s,0\},\mathrm{where}{\varkappa }_1=\sqrt{{\displaystyle \frac{\beta -1}{\beta -4}}}.$$

For the second family $x_0^{\left(2\right)}$ the set of eigenvalues is

$$\mathit{\lambda }=\{\pm i2{\varkappa }_{2}s,\pm i{\varkappa }_{2}s,0\},\mathrm{where}{\varkappa }_2=\sqrt{{\displaystyle \frac{\beta -1}{4\beta -1}}}.$$

Now, the linear transformations that normalize the matrix $M\left({\mathcal{F}}_1^1\right)$ are defined by the matrices

$$Q_1=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ 4{\varkappa }_{1}s& -4{\varkappa }_{1}s& 2{\varkappa }_{3}s& -2{\varkappa }_{3}s& 0& 0\\ {\displaystyle \frac{3\beta s}{4-\beta }}& {\displaystyle \frac{3\beta s}{4-\beta }}& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 2{\varkappa }_1& -2{\varkappa }_1& {\displaystyle \frac{1}{{\varkappa }_1}}& -{\displaystyle \frac{1}{{\varkappa }_1}}& 0& 0\\ 1& 1& 1& 1& 0& 0\end{array}\right),Q_2=\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ {\displaystyle \frac{1}{2}}{\varkappa }_{4}s& -{\displaystyle \frac{1}{2}}{\varkappa }_{4}s& 2{\varkappa }_{2}s& -2{\varkappa }_{2}s& 0& 0\\ 0& 0& {\displaystyle \frac{3\beta s}{4\beta -1}}& {\displaystyle \frac{3\beta s}{4\beta -1}}& 0& 0\\ 0& 0& 0& 0& 1& 0\\ {\displaystyle \frac{1}{2}}{\varkappa }_2& -{\displaystyle \frac{1}{2}}{\varkappa }_2& {\varkappa }_2& -{\varkappa }_2& 0& 0\\ 1& 1& 1& 1& 0& 0\end{array}\right),$$

for the first and second families of solutions, respectively. Here,

$${\varkappa }_3={\displaystyle \frac{2\beta -5}{\sqrt{\left(\beta -4\right)\left(\beta -1\right)}}},{\varkappa }_4={\displaystyle \frac{8\beta -5}{2\sqrt{\left(4\beta -1\right)\left(\beta -1\right)}}}.$$

Applying these transformations to the quadratic part of the EP-system along the family ${\mathcal{F}}_1^1$, we find that all the resonant monomials described in Section 3.4 are absent for all the acceptable values of $\beta$.

Theorem 2.

For the case $A=2B$, $B/C=\beta$, $y_0=z_0=0$, and $1/3⩽\beta ⩽1$, the EP equations are locally integrable in the vicinity of the family ${\mathcal{F}}_1^1$ of the stationary points.

Now, we study the local integrability in the vicinity of SPs of the families ${\mathcal{F}}_2^1$ and ${\mathcal{F}}_3^1$. For each value of $\beta$ from the set of potential integrability (46), we should find the minimal order of resonance that exists for real values of parameters $s,x_0,k$, using Resonant Condition (21), but first let us apply some evident simplifications.

The presence of the resonance $\mathfrak{q}:1$ immediately implies the resonance $1:\mathfrak{q}$. So, the Resonant Condition should be a self-reciprocal polynomial of the value $\mathfrak{q}$, and we can apply the substitution

$$\mathfrak{Q}=\mathfrak{q}+1/\mathfrak{q}$$

(47)

to provide the following simplification. For the family ${\mathcal{F}}_2^1$ Resonant Condition takes the form

$$\begin{array}{c}{Res}_{\mathfrak{q}:1}\left({\mathcal{F}}_2^1\right)=-4{\beta }^2{\left(\beta -1\right)}^2{x}_0^4-4C^2\beta \left(\beta -1\right){\left(2\beta -1\right)}^2\left(\beta \left(2\beta -1\right)\left(3\beta -2\right){\mathfrak{Q}}^2-4{\beta }^2+3\beta -1\right)s^2{x}_0^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hfill +C^4{\left(2\beta -1\right)}^4\left(2{\beta }^2\left(2\beta -1\right)\left(\beta -1\right){\mathfrak{Q}}^2-{\left(4{\beta }^2-3\beta +1\right)}^2\right)s^4=0,\end{array}$$

(48)

and for the family ${\mathcal{F}}_3^1$ Resonant Condition takes the form

$${Res}_{\mathfrak{q}:1}\left({\mathcal{F}}_3^1\right)=4{\left(\beta -1\right)}^2{x}_0^4-4\left(\beta -1\right)\left({\mathfrak{Q}}^2+\beta -3\right){\beta }^2{C}^2{s}^2{x}_0^2+\left(2{\mathfrak{Q}}^2(\beta -1)+{(\beta -3)}^2\right){\beta }^4{C}^4{s}^4=0,$$

(49)

where $\mathfrak{Q}$ is taken from (47) for any rational value of $\mathfrak{q}$. Considering Equations (48) and (49) as bi-quadratic equations, we note that it is not difficult to obtain the minimal values of the resonance order $\mathfrak{q}+1$. The results of these computations are collected in Table 1.

Table 1. Minimal order of a resonance existence for real values of parameters $s,x_0$.

No	1	2	3	4	5	6
$\beta$	1/3	0.6656638025	2/3	0.6668031182	0.6745626725	0.9967129831
${\mathcal{F}}_2^1$	3	–	–	3457	62	3
${\mathcal{F}}_3^1$	3	4	4	4	4	26

According to Table 1, we have the three situations described at the end of the previous subsection.

For Situation 1, we select the values No 1 and No 6 when the resonance of the order 3 exists. The computational procedure here is similar to the procedure described for the family ${\mathcal{F}}_1^1$ above.

Resonant condition (21) takes the form

$$\begin{array}{c}\hfill {Res}_{2:1}\left({\mathcal{F}}_2^1\right)=C^4\left(3{\beta }^2-6\beta +2\right)\left(6{\beta }^2+3\beta -1\right){\left(2\beta -1\right)}^4{s}^4-2C^2\beta \left(\beta -1\right)\left(150{\beta }^3-191{\beta }^2+62\beta -4\right){\left(2\beta -1\right)}^2{s}^2{x}_0^2\\ \hfill -8{\beta }^2{\left(\beta -1\right)}^2{x}_0^4=0.\end{array}$$

(50)

This is a bi-quadratic equation of the parameter $x_0^2$, which must have at least one positive root. Direct substitution $\beta$ from the set $\mathcal{B}$ gives only two cases of the existence of the third-order resonance:

$$\beta ={\displaystyle \frac{1}{3}},\beta \approx 0.9967129831,$$

i.e., for the case (41) (the first value in the set $\mathcal{B}$) and the last value in the set $\mathcal{B}$. So, here we have Situation 1 of Section 7.5.

Resonant condition (50) for $\beta =1/3$ provides the only real solution for $x_0$ and k as follows

$$x_0^2={\displaystyle \frac{1}{6}}{C}^2{s}^2,k^4={\displaystyle \frac{5}{2}}{s}^2.$$

The set of eigenvalues is

$$\mathit{\lambda }=\{\pm 2i\alpha ,\pm i\alpha ,0\},$$

where $\alpha =\sqrt{5s\sqrt{10}}/5$. It makes it possible to compute the matrix Q of the linear normalization transformation $X=QY$ and to apply this transformation to the quadratic part of the EP-equations. After corresponding computations, we see that for Equations (3) and (4) for the system (1) the resonant monomials are absent, but for Equations (1) and (2), their coefficients are equal $\mp i\sqrt{2s\sqrt{10}}/48$. So, they can disappear for the parameter value $s=0$ only, which corresponds to the family ${\mathcal{F}}_1^1$.

According to Hypothesis 2 we do not have local integrability here for case (41).

The second value $\beta \approx 0.9967129831$ (No 6 in Table 1) is an irrational number, and we repeat the above computations not in an analytic way but numerically. We also obtained that the resonant monomials in the first and second equations of the NF have non-zero values.

Theorem 3.

For β values No 1 and No 6 in Table 1, the NFs in the vicinity of the family ${\mathcal{F}}_2^1$ are not integrable.

Now we check the values No 2 and 3 for the family ${\mathcal{F}}_2^1$, where Situation 3 takes place.

As soon as Resonant Condition (48) fails for the real values of the parameters, the EP equations can be reduced to the NF by normalizing transformations. The last step is to prove the convergence of this transformation.

Here we give a detailed description of computations for the value $\beta =2/3$. The characteristic polynomial $\chi \left(\lambda \right)$ takes the form

$$\chi \left(\lambda \right)={\lambda }^6+\left({\displaystyle \frac{7s^2}{8k^2}}+{\displaystyle \frac{9x_0^2}{2C^2{k}^2}}\right){\lambda }^4-{\displaystyle \frac{s^4}{8k^4}}{\lambda }^2.$$

(51)

The parameter k is obtained from the condition (5) and takes the form

$$k^4={\displaystyle \frac{C^2{s}^2+9x_0^2}{C^2}}\Rightarrow k^2=\pm \sqrt{C^2{s}^2+9x_0^2}/C.$$

So, Equation (51) can be rewritten after canceling by ${\lambda }^2$ as follows

$$\tilde{\chi }\left(\lambda \right)=\left(8C^2{s}^2+72x_0^2\right){\lambda }^4\pm {\displaystyle \frac{1}{C}}\sqrt{C^2{s}^2+9x_0^2}\left(7C^2{s}^2+36x_0^2\right){\lambda }^2-s^4{C}^2.$$

For real values of all parameters $C,s,x_0$ we find that its discriminant relative to ${\lambda }^2$ is always positive:

$${\displaystyle \frac{9\left(C^2{s}^2+9x_0^2\right)\left(3C^2{s}^2-4Cs{x}_0+12x^2\right)\left(3C^2{s}^2+4Cs{x}_0+12x^2\right)}{C^2}}>0,$$

and the free term of $\tilde{\chi }\left(\lambda \right)$ is always negative. As a consequence, we obtain two pairs of non-zero eigenvalues: a pair of real and a pair of purely imaginary. According to Moser’s theorem [12] the normalizing transformation is convergent.

For the value No 2, we can also provide similar computations and we obtained the same result.

Theorem 4.

For β values No 2 and No 3 in Table 1, the corresponding NFs in the vicinity of the family ${\mathcal{F}}_2^1$ are integrable.

Finally, we give the description of the results of the NF computation for the family ${\mathcal{F}}_3^1$ for values No 2, 3, 4, 5, i.e., when Situation 2 takes place with resonance of order 4.

For the fixed value of ${\beta }^{*}$ from the set $\mathcal{B}$ (46), the normalization procedure consists of the following steps.

Step 1: 

Using the resonant condition (49) compute the real value of parameter $x_0$, which provides the existence of the fourth order resonance. After that, compute the corresponding value of the parameter k such that ${\gamma }_1^2+{\gamma }_2^2=1$, where ${\gamma }_j$ satisfies the family ${\mathcal{F}}_3^1$ conditions (28).

Step 2: 

Substitute all the parameters values in the matrix M of the linear part of the system (1) and obtain matrix $\tilde{M}(s,{\beta }^{*})$. Now we can obtain the matrix Q of a linear transformation $X=QY$, and rewrite the EP equations in the form

$$Y^{\prime }=JY+N_2\left(Y\right),J=diag({\lambda }_1,-{\lambda }_1,3{\lambda }_1,-3{\lambda }_1,0,0),$$

(52)

and $N_2\left(Y\right)$ is a vector of quadratic forms.

Step 3: 

Now apply a standard normalization technique (see, e.g., [13]), providing a transformation $Y=Z+H_1\left(Z\right)+H_2\left(Z\right)+\dots$, which reduces the system (52) into the NF $Z^{\prime }=JZ+G_1\left(Z\right)+\dots$. For the resonance of order 4, it is necessary to make one single step of normalization by solving the so-called homological equation

$$G_1\left(Z\right)+\mathcal{L}\left(H_1\left(Z\right)\right)=N_2\left(Z\right),$$

(53)

where $\mathcal{L}$ is a Lie bracket $\mathcal{L}\left(H_k\left(Z\right)\right)=D{H}_k\left(Z\right)JZ-J{H}_k\left(Z\right)$. Solving Equation (53), we obtain the terms of the second-order NF $G_1\left(Z\right)$ and second order transformation terms $H_1\left(Z\right)$. These terms are enough to construct the right-hand side of the next order homological equation and to check the coefficients $c_{ijk}$ of the resonant monomials, where i is the number of the equation, $j,k$ are indices of the variables Z. For 4th order resonance the resonant monomials are the following:

$$c_{123}{Z}_2^2{Z}_3,c_{214}{Z}_1^2{Z}_4,c_{310}{Z}_1^3,c_{420}{Z}_2^3.$$

For the value $\beta =2/3$ (No 3 in Table 1), the coefficients of the resonant terms can be computed symbolically, but for other values (No 2, 4, 5) only numerically. The coefficients obtained, divided by $\sqrt{s}$, are presented in Table 2.

Table 2. Table of the resonant terms coefficients for potentially integrable cases. Here $\delta =i{42}^{3/4}\sqrt{6}/72$.

No	${\mathit{c}}_{123}$	${\mathit{c}}_{214}$	${\mathit{c}}_{310}$	${\mathit{c}}_{420}$
2	$-0.192267822892i$	$0.192267822892i$	$0.0413172384893i$	$-0.0413172384893i$
3	$-\delta$	$\delta$	$\delta /3$	$-\delta /3$
4	$-0.192107258655$	$-0.192107258655$	$-0.0413243675971$	$-0.0413243675971$
5	$-0.190226341400$	$-0.190226341400$	$-0.0411825670879$	$-0.0411825670879$

Theorem 5.

For each value β No 2, 3, 4, 5 in Table 1 of the family ${\mathcal{F}}_3^1$, the NF has non-zero resonant monomials and therefore is not locally integrable.

So, now we can summarize all the computations for the case $\mathsf{\ell }=1$.

Theorem 6.

All known cases of global integrability were obtained. For all new computed potentially integrable cases of the form $A=2B$, $B=\beta C$, with $\beta \in \mathcal{B}$, there are no global integrability.

8. Results for Case $\mathsf{\ell }=\mathbf{2}$

Now consider the case $\mathsf{\ell }=2$, that is, $x_0,y_0\ne 0,z_0=0$, so the set of non-trivial parameters is ${\Xi }_2=\{x_0,y_0,A,B,C\}$. Then the EP equations have two families of stationary points: family ${\mathcal{F}}_1^2$ (below we repeat its representation (30) again)

$${\mathcal{F}}_1^2:\left\{p={\displaystyle \frac{x_0}{k(C-A)}},q={\displaystyle \frac{y_0}{k(C-B)}},r=s,{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3={\displaystyle \frac{r}{k}},\right\},$$

where ${\gamma }_1^2+{\gamma }_2^2+{\gamma }_3^2=1$, $A\ne C\ne B$, and family ${\mathcal{F}}_2^2$ (below we repeat its representation (31) again)

$${\mathcal{F}}_2^2:\left\{p=-{\displaystyle \frac{x_0}{k(A+s)}},q=-{\displaystyle \frac{y_0}{k(B+s)}},r=0,{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3=0,{\gamma }_1^2+{\gamma }_2^2=1\right\};$$

s is a parameter. For the family ${\mathcal{F}}_2^2$ parameter $s\ne 2C$.

8.1. Conditions of Potential Integrability for Family ${\mathcal{F}}_1^2$

The coefficients of the characteristic polynomial $\chi \left({\mathcal{F}}_1^2\right)$ are the following:

$$\begin{array}{cc}\hfill a_4=& {\displaystyle \frac{\left(2AB-(A+B)C+C^2\right)s^2}{BA{k}^2}}+{\displaystyle \frac{\left({(A-C)}^2-(2A-3C)B\right)x_0^2}{C{\left(A-C\right)}^{2}B{k}^2}}-{\displaystyle \frac{\left((2B-3C)A-{(B-C)}^2\right)y_0^2}{CA{\left(B-C\right)}^2{k}^2}},\hfill \\ \hfill a_2=& {\displaystyle \frac{\left(A-C\right)\left(B-C\right)s^4}{AB{k}^4}}-{\displaystyle \frac{\left(B-C\right)\left(3A-4C\right)s^2{x}_0^2}{CA\left(A-C\right)B{k}^4}}-{\displaystyle \frac{\left(3B-4C\right)\left(A-C\right)s^2{y}_0^2}{\left(B-C\right)BAC{k}^4}}.\hfill \end{array}$$

Along this family the parameter k should satisfy the following relation:

$${\left(A-C\right)}^2{\left(B-C\right)}^2\left(k^4+s^2\right)=x_0^2{(B-C)}^2+y_0^2{(A-C)}^2.$$

(54)

For this family, and according to Step 2, we compute the numerator G of the discriminant $D_{\lambda }\left(\chi \right)$:

$$G\left({\mathcal{F}}_1^2\right)=C^2{g}_4^2{s}^4+2g_4{g}_2{s}^2+g_0^2,$$

(55)

where

$$\begin{array}{cc}\hfill g_4=& C^2{\left(B-C\right)}^2{\left(A-C\right)}^2{\left(A+B-C\right)}^2,\hfill \\ \hfill g_2=& A{\left(B-C\right)}^2\left(2A^{2}B-A^{2}C-6ABC+2A{C}^2+5B{C}^2-C^3\right)x_0^2\hfill \\ \hfill & +B{\left(A-C\right)}^2\left(2A{B}^2-6ABC+5A{C}^2-B^{2}C+2B{C}^2-C^3\right)y_0^2,\hfill \\ \hfill g_0=& A{\left(B-C\right)}^2\left(A^2-2AB-2AC+3CB+C^2\right)x_0^2-B{\left(A-C\right)}^2\left(2AB-3AC-B^2+2CB-C^2\right)y_0^2.\hfill \end{array}$$

For the family ${\mathcal{F}}_1^2$, the secondary discriminant ${\Delta }_s$ over $s^2$ is the following

$$\begin{array}{cc}\hfill {\Delta }_{s^2}\left({\mathcal{F}}_1^2\right)& \cong AB{C}^2{\left(B-C\right)}^6{\left(A-C\right)}^6{\left(A+B-C\right)}^2{f}_{11}{f}_{12},\mathrm{where}\hfill \\ \hfill f_{11}& \stackrel{\mathrm{def}}{=} A\left(B-C\right)x_0^2+B\left(A-C\right)y_0^2,\hfill \\ \hfill f_{12}& \stackrel{\mathrm{def}}{=} {\left(B-C\right)}^2{\left(A-2C\right)}^2{x}_0^2+{\left(B-2C\right)}^2{\left(A-C\right)}^2{y}_0^2,\hfill \end{array}$$

(56)

Other secondary discriminants can be given in a brief description only because of their awkwardness.

$${\Delta }_A\left({\mathcal{F}}_1^2\right)\cong C^{17}{(B-C)}^{27}{B}^8{s}^6{x}_0^{14}{g}_{1A},$$

where $g_{1A}$ is a polynomial of degree 7 of $s^2$ with coefficients in the form of homogeneous polynomials of parameters $B,C,x_0,y_0$. So, it is omitted.

In terms of secondary discriminants ${\Delta }_A$ and ${\Delta }_B$ symmetrical to each other under the transformation $A\leftrightarrow B$, $x_0\leftrightarrow y_0$ one obtains

$${\Delta }_B\cong C^{17}{(A-C)}^{27}{A}^8{s}^6{y}_0^{14}{g}_{1B},$$

where the polynomial $g_{1B}$ is similar to $g_{1A}$ and is also omitted.

$${\Delta }_C\left({\mathcal{F}}_1^2\right)\cong {(A-B)}^{22}{A}^{26}{B}^{26}{x}^{14}{y}^{14}{s}^{30}{g}_{1C},$$

where $g_{3C}$ is a quasi-homogeneous polynomial of parameters $s,x_0,y_0$ of degree 46 consisting of 262 monomials. It is omitted here.

Finally, we obtained for the family ${\mathcal{F}}_1^2$ the following factors

$$A-B,A-C,B-C,A+B-C,f_{11},f_{12},$$

(57)

for which it is possible to provide further computations.

8.2. Conditions of Potential Integrability for Family ${\mathcal{F}}_2^2$

The coefficients of the characteristic polynomial $\chi \left({\mathcal{F}}_2^2\right)$ are the following:

$$\begin{array}{cc}\hfill a_4=& {\displaystyle \frac{\left(A^2-2AB-2AC+2BC-sB-sC\right)x_0^2}{B{\left(A+s\right)}^2{k}^{2}C}}-{\displaystyle \frac{\left(2AB-2AC+sA-B^2+2BC+sC\right)y_0^2}{C{k}^2{\left(B+s\right)}^{2}A}},\hfill \\ \hfill a_2=& {\displaystyle \frac{\left(C+s\right)\left(B+s\right)x_0^4}{C{\left(A+s\right)}^4{k}^{4}B}}-{\displaystyle \frac{\left(3A^2-8AB-sA+3B^2-sB\right)\left(C+s\right)x_0^2{y}_0^2}{C{\left(A+s\right)}^2{\left(B+s\right)}^2{k}^{4}AB}}+{\displaystyle \frac{\left(C+s\right)\left(A+s\right)y_0^4}{C{\left(B+s\right)}^4{k}^{4}A}}.\hfill \end{array}$$

For this family, and according to the Step 2, we compute the numerator $G\left({\mathcal{F}}_2^2\right)$ of the discriminant $D_{\lambda }\left(\chi \right)$, which is a huge polynomial of degree 6 with coefficients in the form of quasi-homogeneous polynomials of the parameters $A,B,C,x_0,y_0$. Due to its awkwardness, it is omitted. However, we managed to compute all its secondary discriminants ${\Delta }_{\xi }$ for $\xi \in \{s,A,B,C\}$.

$${\Delta }_s\left({\mathcal{F}}_2^2\right)\cong A^8{B}^8{C}^3{x}_0^{12}{y}_0^{12}{\left(A-B\right)}^{16}{\left(C+A-B\right)}^2{\left(A-B-C\right)}^2{f}_{21}^2{f}_{22},$$

where

$$f_{21}=A{\left(A-2B\right)}^2\left(C+A-B\right)x_0^2-B{\left(2A-B\right)}^2\left(A-B-C\right)y_0^2,$$

and $f_{22}$ is a homogeneous polynomial of degree 12 depending on the even degrees of $x_0,y_0$ only. It is also omitted.

$${\Delta }_A\left({\mathcal{F}}_2^2\right)\cong B^6{s}^6{y}^4{C}^3{x}_0^{12}{\left(x_0^2+y_0^2\right)}^2{\left(C+s\right)}^3{\left(B+s\right)}^{18}{\left(B+C+s\right)}^2{g}_{1A}{g}_{2A}^2,$$

where $g_{1A}$ is a huge quasi-homogeneous polynomial of degree 12 of parameters $x_0,y_0$ and is omitted here whereas

$$\begin{array}{c}{g}_{2A}={\left(B+s\right)}^3\left(B-C\right)\left(B^2+BC-Bs+3Cs\right)x_0^4\hfill \\ \hspace{1em} -B\left(B+s\right)\left(4B^4+B^{3}C+B^{3}s-3B^2{C}^2+12B^{2}Cs-B^2{s}^2-12B{C}^{2}s+9BC{s}^2+B{s}^3-10C^2{s}^2-2C{s}^3\right)y_0^2{x}_0^2\\ \hfill +B^2{\left(B+2s\right)}^2\left(B-2C\right)\left(B+C+s\right)y_0^4,\end{array}$$

Secondary discriminant ${\Delta }_B\left({\mathcal{F}}_2^2\right)$ is constructed by the symmetry transformation $A\leftrightarrow B$, $x_0\leftrightarrow y_0$ and it is omitted here.

Finally,

$${\Delta }_C\left({\mathcal{F}}_2^2\right)\cong {\left(B+s\right)}^2{\left(A+s\right)}^{2}AB{g}_{1C}{g}_{2C}^2,$$

where

$$\begin{array}{cc}\hfill g_{1C}=& A{\left(B+s\right)}^5{x}^4-{\left(B+s\right)}^2{\left(A+s\right)}^2\left(3A^2-8AB-As+3B^2-Bs\right)x^2{y}^2+B{\left(A+s\right)}^5{y}^4,\hfill \\ \hfill g_{2C}=& A\left(B+s\right)\left(A-2B\right)x^2-B\left(A+s\right)\left(2A-B\right)y^2.\hfill \end{array}$$

Collecting together computations for the family ${\mathcal{F}}_2^2$ there are the following non-trivial factors except $s+A$ and $s+B$:

$$A-B,C+A-B,A-B-C,f_{21},B+C+s,A+C+s,g_{1A}(A,B),g_{1B},g_{1C},g_{2C}.$$

(58)

Comparing the sets of factors for the first (57) and for the second (58) families, one can note that potentially integrable cases are

1.

$A=B$;

2.

Roots of any pair of other factors: one from the equation (57) and another from (58), totally $6\times 4=24$ cases.

For the family ${\mathcal{F}}_2^2$ we also have factor $A-B$. So, both mentioned above cases are integrable. Other cases are not evident. But the case $A=B$ is not potentially integrable because $f_{12}=0$ in the two cases mentioned above only and not in the case $A=B$.

For polynomials of the families ${\mathcal{F}}_1^2$ and ${\mathcal{F}}_2^2$, one can obtain pairs of polynomials depending on $A,B,C$ only in the following way. For ${\mathcal{F}}_1^2$ from the polynomials ${\Delta }_A$, ${\Delta }_B$, and ${\Delta }_C$ eliminate the parameter $x_0$ by computing their resultants with $f_{11}$ in $x_0$ and then, from the polynomials obtained taking them in pairs, remove the parameter s. Similarly for the family ${\mathcal{F}}_2^2$ from the polynomials $f_{22}$, ${\Delta }_A$, ${\Delta }_B$ and $g_{1C}$ one can eliminate the parameter $x_0$ using $g_{2C}$. Now, one obtains three polynomials of $s,A,B,C$ and then eliminates the parameter s from the pair ${\Delta }_A$ and ${\Delta }_B$ using the polynomial $g_{1C}$. After such eliminations four pairs of polynomials depending on $A,B,C$ only are studied to obtain their common roots. These roots would be potentially integrable cases. Such computations deal with huge polynomials, which demand either powerful hardware or sophisticated software for their implementation. At this point, the authors are not ready for this.

9. Case $\mathsf{\ell }=\mathbf{3}$

In this case the EP equations (1) has the only family (32) of SP

$${\mathcal{F}}_1^3:\left\{p=-{\displaystyle \frac{x_0}{k(A+s)}},q=-{\displaystyle \frac{y_0}{k(B+s)}},r=-{\displaystyle \frac{z_0}{k(C+s)}},{\gamma }_1={\displaystyle \frac{p}{k}},{\gamma }_2={\displaystyle \frac{q}{k}},{\gamma }_3={\displaystyle \frac{r}{k}}\right\},$$

for which any permutation from Section 2 is an automorphism. The numerator $G\left({\mathcal{F}}_1^3\right)$ has the following form:

$$\begin{array}{cc}\hfill G\left({\mathcal{F}}_1^3\right)=& A^2{\left(B+s\right)}^4{\left(C+s\right)}^4\left({\left(B-C\right)}^2{\left(A+s\right)}^2-2\left(AB+AC-4BC\right)\left(A-B-C\right)\left(A+s\right)+A^2{\left(A-B-C\right)}^2\right)x_0^4\hfill \\ \hfill & -2A{\left(A+s\right)}^{2}B{\left(B+s\right)}^2{\left(C+s\right)}^4\left(-\left(B-C\right)\left(A-C\right)\left(A+s\right)\left(B+s\right)-\left(2AC+B^2-5BC\right)\left(A-B+C\right)\left(A+s\right)\right.\hfill \\ \hfill & \left.+\left(A^2-5AC+2BC\right)\left(A-B-C\right)\left(B+s\right)+AB\left(A-B+C\right)\left(A-B-C\right)\right)x_0^2{y}_0^2\hfill \\ \hfill & -2A{\left(A+s\right)}^2{\left(B+s\right)}^{4}C{\left(C+s\right)}^2\left(\left(B-C\right)\left(A-B\right)\left(A+s\right)\left(C+s\right)-\left(A+B-C\right)\left(2AB-5BC+C^2\right)\left(A+s\right)\right.\hfill \\ \hfill & \left.+\left(A^2-5AB+2BC\right)\left(A-B-C\right)\left(C+s\right)+AC\left(A+B-C\right)\left(A-B-C\right)\right)x_0^2{z}_0^2\hfill \\ \hfill & +{\left(A+s\right)}^4{B}^2{\left(C+s\right)}^4\left({\left(A-C\right)}^2{\left(B+s\right)}^2+2\left(AB-4AC+BC\right)\left(A-B+C\right)\left(B+s\right)+B^2{\left(A-B+C\right)}^2\right)y_0^4\hfill \\ \hfill & +2{\left(A+s\right)}^{4}B{\left(B+s\right)}^{2}C{\left(C+s\right)}^2\left(\left(A-C\right)\left(A-B\right)\left(B+s\right)\left(C+s\right)+\left(A+B-C\right)\left(2AB-5AC+C^2\right)\left(B+s\right)\right.\hfill \\ \hfill & \left.-\left(5AB-2AC-B^2\right)\left(A-B+C\right)\left(C+s\right)+BC\left(A+B-C\right)\left(A-B+C\right)\right)y_0^2{z}_0^2\hfill \\ \hfill & -{\left(A+s\right)}^4{\left(B+s\right)}^4{C}^2\left(-{\left(A-B\right)}^2{\left(C+s\right)}^2+2\left(A+B-C\right)\left(4AB-AC-BC\right)\left(C+s\right)-C^2{\left(A+B-C\right)}^2\right)z_0^4.\hfill \end{array}$$

It can be considered as a quasi-homogeneous polynomial of degree 10 over the parameter s or a quasi-homogeneous polynomial of degree 6 over one of the parameters A, B or C. The secondary discriminant ${\Delta }_s\left(G\right)$ is a polynomial of 133,881 monomials of the remaining parameters $A,B,C,x_0,y_0,z_0$. The authors attempted to factorize it and to find smaller factors failed. Computations of secondary discriminants ${\Delta }_{\xi }\left(G\right)$, $\xi \in \{A,B,C\}$ were also not very successful, e.g.,

$${\Delta }_A\left(G\right)=B^3{C}^3{\left(C+s\right)}^{18}{\left(B+s\right)}^{18}{\left(B+C+s\right)}^2{s}^6{x}_0^{12}{g}_{1A}(B,C,s,x_0,y_0,z_0),$$

where $g_{1A}$ is a polynomial consisting of 127,212 monomials. Its factorization failed. We tried to consider some particular subcases $A=B$, $A=2B$, $A=3B$, but none of these substitutions could give us any simplification.

Anyway, the roots of ${\Delta }_s\left(G\right)$ form a 5-dimensional algebraic variety of potentially integrable cases in the 6-dimensional space of parameters. Of course, not all of them are globally integrable, but this variety may be rather rich. Additional integrable cases can be obtained from the roots of other secondary discriminants ${\Delta }_A$, ${\Delta }_B$, and ${\Delta }_C$.

10. Conclusions

For $\mathsf{\ell }=1$, we obtained all cases of potential integrability, particularly all known cases of global integrability, and six new cases of the form $A=2B$, $B=\beta C$, where $\beta$ takes values from the set $\mathcal{B}$ (46). None of these six new cases is locally integrable as soon as the NF of the EP equations contains non-zero resonant monomials either in the vicinity of the family ${\mathcal{F}}_2^1$ or the family ${\mathcal{F}}_3^1$.

For $\mathsf{\ell }=2$ we obtained all known cases of global integrability and several new cases of potential integrability.

For $\mathsf{\ell }=1$ and $\mathsf{\ell }=2$ there are discrete sets of integrable cases. But for $\mathsf{\ell }=3$ they can form continued sets in the space of parameters. In the book [14] there are several problems where the method presented in the article may be useful.