Exact Solutions of Maxwell Vacuum Equations in Petrov Homogeneous Non-Null Spaces

Abstract

The classification of exact solutions of Maxwell vacuum equations for pseudo-Riemannian spaces with spatial symmetry (homogeneous non-null spaces in Petrov) in the presence of electromagnetic fields invariant with respect to the action of the group of space motions is summarized. A new classification method is used, common to all homogeneous zero spaces of Petrov. The method is based on the use of canonical reper vectors and on the use of a new approach to the systematization of solutions. The classification results are presented in a form more convenient for further use. Using the previously made refinement of the classification of Petrov spaces, the classification of exact solutions of Maxwell vacuum equations for spaces with the group of motions $G_3\left(VIII\right)$ is completed.

1. Introduction

Of particular interest in the theory of gravity and electromagnetism are spaces and fields endowed with symmetry. The symmetry of space is manifested through the symmetry of geodesic lines and is defined by vector or tensor Killing fields. In addition, this symmetry is manifested in physical fields through the motion integrals of such equations as the Hamilton–Jacobi, Klein–Gordon–Fock, Dirac–Fock, etc., equations.

In gravity theory, the most interesting spaces are those with Lorentz signature that admit three Killing fields. These spaces include two types of spaces: Stäckel spaces (see [1,2,3,4,5,6,7,8,9]) and Petrov spaces (see [10,11,12]). Both types of the spaces have complete sets consisting of three Killing fields. For Stäckel spaces, the complete sets consist of mutually commuting Killing tensor fields of the first rank (vector fields) and/or of the second rank. For Petrov spaces, the complete sets consist of three Killing vector fields. Such sets allow one to introduce a privileged coordinate system in which the components of the metric tensor contain functions, each of which depends on only one of the variables.

This makes it possible to solve the separation of variables problem (including in the presence of physical fields) in the field equations and in the motion equations for a single test particle.

The above-mentioned feature of this type of space allows these equations to be reduced to systems of ordinary differential equations. All the exact solutions of motion equations known to date were found thanks to precisely this circumstance. For Stäckel spaces (see [13,14,15,16,17,18,19]). For Petrov spaces, see [10,12,20,21,22,23,24,25,26]. Among these solutions are such important solutions as the solutions [27,28,29,30,31,32,33,34].

Stackel spaces continue to attract the attention of researchers, since they are used as a basis for constructing models of real processes occurring in the presence of a gravitational field. Among them, metrics with spherical or axial symmetry are most often considered, including in multidimensional theories of gravity. As an example, we can cite the papers [35,36,37,38,39,40,41,42,43]. As examples of the study of Petrov spaces in cosmology, one can cite recent papers: [44] (where the Hamiltonian approach to cosmological models according to Bianchi is developed) and [45] (authors claim to have obtained an ideal characterization of spatially homogeneous cosmologies).

One of the most important examples of metrics related to the class of Stäckel spaces are the De Sitter, Anti-de Sitter, Friedman, and some other metrics that underlie the vast majority of cosmological models, including models of the modified theory of gravity. A huge number of articles are devoted to these models (see, for example, [46,47,48,49,50,51,52,53,54,55,56,57,58]).

Let us note that recently wave spaces have attracted considerable interest. The manifold of such spaces intersects with the manifold of null Stäckel spaces, which have null Killing vector fields in their complete sets. The general form of the metric of a null Stäckel space was first found by V.N. Shapovalov. Therefore, in paper [59], it was proposed to call null Stäckel spaces with null Killing vector field Shapovalov wave spaces. As an example of works devoted to various aspects of Shapovalov wave spaces, one can cite papers [60,61,62,63].

In Petrov spaces, another method of exact integration of the equations of motion of a test particle is used. It is based on the use of a non-commutative group of motions of the space. Therefore, it is called the method of non-commutative integration. (see article [20]). Note that a large number of articles are devoted to the consideration of other problems in Petrov spaces (see articles [64,65,66,67,68,69]) and to the classification of homogeneous Stäckel spaces (see [70,71,72]).

An important direction is the consideration of problems of symmetry theory in homogeneous spaces (see, for example, [73,74,75]) too.

In conclusion of this section, let us note the following.

The results obtained after applying the theory of symmetry (including the theory of continuous groups of transformations) to the equations of mathematical physics make it possible to list all nonequivalent solutions of these equations that are invariant with respect to the corresponding symmetry operators. This is the purpose of the classification. To date, classifications have been constructed (completely or partially) for the Laplace equations, d’Alembert equations, for single-particle equations of motion, etc. After the appearance of general relativity, the most important direction of classification became the classification of Riemannian and pseudo-Riemannian spaces with a given symmetry (the problem has been completely solved for Stackel spaces and for the homogeneous Petrov spaces), as well as the classification of mathematical physics equation solutions in these spaces. In addition, Einstein’s equations were added to the number of basic equations of mathematical physics. A large number of papers are devoted to solving classification problems in general relativity. From the point of view of mathematical physics, completely solved classification problems are of particular interest.

The range of such solved problems is expanded by the problem that has been solved in this article. The classification of exact solutions to Maxwell vacuum equations in homogeneous nonzero Petrov spaces with an invariant electromagnetic field, begun in papers [76,77,78,79,80] is completed. A complete list of such solutions is obtained.

2. Non-Null Petrov Spaces

Consider a Riemannian space $V_4$ with a Lorentz signature and with a group of motions $G_3\left(N\right)$ acting simply transitively on the hypersurface $V_3$. Here N corresponds to the number of the group $G_3$ according to Bianchi classification. The hypersurface $V_3$ has the geometry of a three-dimensional homogeneous space $V_3\left(N\right)$. The space $V_4$ is called a homogeneous Petrov space and is denoted by $V_4\left(N\right)$. Petrov was the first who carried out a complete classification of the spaces $V_4\left(N\right)$ in papers [10,11]. In the case where the hypersurfaces of transitivity $V_3$ are non-null spaces with Euclidean signature, $V_4\left(N\right)$ are also denoted as homogeneous spaces of type N according to the Bianchi classification.

Throughout the text, the following notations for indices are used:

$$i,j,k,r÷1,2,3,0;\alpha ,\beta ,\gamma ,\delta ...÷1,2,3;a,b,c,d÷1,2,3.$$

The fourth coordinate of the holonomic coordinate system $\left\{u^i\right\}$ is supplied with the index $i=0$ ($u^0$), and the numbering of coordinate indices always starts with one and ends with zero (when indices $i,j,k,l$ are used). Greek letters denote the coordinate indices of the semi-geodetic coordinate system $\left\{u^{\alpha }\right\}$, related to the transitivity hypersurface of the group $G_3\left(N\right)$. Latin letters a, b, c, and d denote the indices of the non-holonomic coordinate system associated with the group $G_3\left(N\right)$, as well as the indices of the structural constants and the numbers of the reper vectors.

The Petrov method of classification of non-null spaces $V_4\left(N\right)$ is as follows:

• A semi-geodetic coordinate system, in which the metric tensor of the space $V_4\left(N\right)$ has the form:

  $$g^{ij}=\left(\begin{array}{cccc}{g}^{\alpha \beta }\left(u^i\right)& & & 0\\ & & & 0\\ & & & 0\\ 0& 0& 0& -\epsilon {\tau }^2\left(u^0\right)\end{array}\right),det∥g_{\alpha \beta }∥=\epsilon G{\left(u^0\right)}^2,\epsilon =\pm 1,$$

  (1)

  is introduced. Unless otherwise stated, we believe $\tau =1.$ Obviously, admissible coordinate transformations:

  $${\tilde{u}}^{\alpha }={\tilde{u}}^{\alpha }\left(u^{\beta }\right),{\tilde{u}}^0={\tilde{u}}^0\left(u^0\right)$$

  (2)

  do not violate the form (1) of the metric tensor.
• The equations of the structure:

  $$\left[X_a,X_b\right]=C_{ab}^c{X}_c(X_a={\xi }_a^i{p}_i={\xi }_a^i{\partial }_i)$$

  (3)

  are integrated, and the Killing vector fields ${\xi }_a^i$ are found. Using admissible coordinate transformations (2), arbitrariness in the obtained solutions is eliminated.
• Using the functions ${\xi }_a^{\alpha }$, the components of the tensor $g^{\alpha \beta }$ can be found by integrating the Killing equations:

  $$g_{,\gamma }^{\alpha \beta }{\xi }_a^{\gamma }=g^{\alpha \gamma }{\xi }_{a,\gamma }^{\beta }+g^{\beta \gamma }{\xi }_{a,\gamma }^{\alpha }.$$

  (4)
  As is known, the metric tensor of the non-null homogeneous Petrov space can also be defined using a triad of vectors of the canonical frame ${\zeta }_a^{\alpha }$, satisfying the equations of the structure:

  $$\left[Y_a,Y_b\right]=C_{ab}^c{Y}_c,(Y_a={\zeta }_a^i{p}_i).$$

  (5)
  In the same coordinate system the vector fields ${\xi }_a^{\alpha },{\zeta }_a^{\alpha }$ are interconnected by equations

  $${\zeta }_{a,\beta }^{\alpha }={\xi }_{\beta }^b{\xi }_{b,\gamma }^{\alpha }{\zeta }_a^{\gamma }\Rightarrow {\xi }_{a,\beta }^{\alpha }={\zeta }_{\beta }^b{\zeta }_{b,\gamma }^{\alpha }{\xi }_a^{\gamma },$$

  (6)

  where ${\xi }_{\alpha }^b,{\zeta }_{\alpha }^b$ are covariant components of the Killing vectors and the canonical frame, respectively:

  $${\xi }_{\beta }^a{\xi }_a^{\alpha }={\zeta }_{\beta }^b{\zeta }_b^{\alpha }={\delta }_{\beta }^{\alpha },{\xi }_{\alpha }^a{\xi }_b^{\alpha }={\zeta }_{\alpha }^a{\zeta }_b^{\alpha }={\delta }_b^a.$$
  (see [81] (p. 484)) Therefore, the classification by the Petrov method should be supplemented by a final stage:
• Classification of all independent solutions of the Equation (6).

This stage is performed in the paper [12]. The integration of the equation of the structure (3) in the semi-geodetic coordinate system (1) has been carried out by A. Z. Petrov in the book [10] (see further comments in Section 5 for the group of motions $G_3\left(VIII\right)$). The sets of vector fields obtained by Petrov were considered as sets of Killing vector fields ${\xi }_a^{\alpha }$. However, they can also be considered as triads of vector fields ${\zeta }_a^{\alpha }$ of the canonical frame. In this case, one obtains an alternative classification method. The metric tensor on the hypersurface $V_3\left(N\right)$ can be immediately found in the form:

$$g_{\alpha \beta }={{\zeta }_{\alpha }}^a{{\zeta }_{\beta }}^b{\eta }^{ab}\left(u^0\right)\Rightarrow G=\zeta \eta ({\eta }^2=\epsilon det∥{\eta }_{\alpha \beta }∥,\zeta =det∥{\zeta }_a^{\alpha }∥).$$

(7)

When studying physical processes occurring in Petrov spaces, it is necessary to know the integrals of motion of the geodesic equations. In Petrov spaces, there are at least three integrals of motion linear in momenta. They are defined by Killing vector fields. Therefore, the classification of solutions to the system of Equation (6) is the final stage in this method of classifying Petrov spaces. This stage will allow one to find the Killing vector fields ${\xi }_a^{\alpha }$ in the same coordinate system.

Let us note that the fields ${\zeta }_a^{\alpha }$ and ${\xi }_a^{\alpha }$ can be swapped. In this case, the coordinate transformation ${\tilde{u}}^{\alpha }={\tilde{u}}^{\alpha }\left(u^{\beta }\right)$ should be carried out according to the formulas:

$${\tilde{\xi }}_a^{\alpha }\left(\tilde{u}\right)={\zeta }_a^{\alpha }\left(\tilde{u}\right),{\tilde{\zeta }}_a^{\alpha }\left(\tilde{u}\right)={\xi }_a^{\alpha }\left(\tilde{u}\right).$$

(8)

The new coordinate system $\left\{u^i\right\}$ can be found from the system of differential equations:

$${\xi }_a^{\alpha }\left(u\right)={\displaystyle \frac{\partial {\tilde{u}}^{\alpha }}{\partial u^{\beta }}}{\zeta }_a^{\beta }\left(u\right).$$

(9)

However, the solution to this problem has no practical significance.

Obviously, when the Petrov method is supplemented by the fourth stage, both methods give the same results presented in different coordinate systems. An alternative approach is more convenient when the classification results are used to integrate field equations with physical fields (for example, with electromagnetic fields) invariant with respect to the action of the group of motions $G_3\left(N\right)$. In the present paper, this is demonstrated on the example of Maxwell vacuum equations for an electromagnetic field invariant under the action of the group $G_3\left(N\right)$. The transition to a non-holonomic canonical frame allows one to reduce Maxwell’s equations to a system of algebraic equations. Note that the classification problem was considered within the framework of Petrov’s method in [76,77,78,80,80]. Some of the solutions obtained there are presented in quadratures. Using the new method, one has obtained all solutions in explicit form. The method is common to all homogeneous non-null Petrov spaces.

The following problems are solved in the paper.

• All solutions of Maxwell vacuum equations have been obtained in explicit form and classified. The common solution scheme have been used for all groups of motions, which made it possible to significantly simplify the solution procedure and further simplify and systematize the previously obtained solutions.
• The classification of exact solutions of Maxwell vacuum equations for Petrov spaces with an unsolvable group of motions $G_3\left(VIII\right)$ has been supplemented by the classification for two new Petrov spaces.
• Section 6 provides a complete list of the solutions obtained.

3. Maxwell Vacuum Equations in the Canonical Frame

As it has been shown in the works [26], in the space $V_4\left(N\right)$ the Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged particle have integrals of motion that are linear in momenta if the external electromagnetic field is invariant with respect to the group $G_r\left(N\right)$ acting simply transitively on the non-null hypersurface $V_3\left(N\right)$. Having chosen the gauge of the electromagnetic potential in the form: $A_0=0,$ one obtains the following condition on the electromagnetic potential:

$${\left({\xi }_a^{\alpha }{A}_{\alpha }\right)}_{,\beta }{\xi }_b^{\beta }=C_{ba}^c{A}_{\alpha }{\xi }_c^{\alpha }.$$

(10)

Let us denote the non-holonomic components of the electromagnetic field vector in the frame ${\xi }_b^{\alpha }$ as follows:

$${\mathbf{A}}_a={\xi }_a^{\alpha }{A}_{\alpha }.$$

Then the Equation (10) takes the form:

$${\mathbf{A}}_{a/b}=C_{ba}^c{\mathbf{A}}_c.$$

(11)

Throughout the text the following notations are used:

$${\xi }_a^{\alpha }{F}_{,\alpha }=F_{/a},{\zeta }_a^{\alpha }{F}_{,\alpha }=F_{|a}.$$

Next notation

$${\mathbb{A}}_a={\zeta }_a^{\alpha }{A}_{\alpha }$$

(12)

is used for the non-holonomic components of the of the electromagnetic field vector potential in the canonical frame. Throughout the text, functions denoted by lowercase Greek letters depend only on the variable $u^0$.

Lemma 1.

The non-holonomic components (12) of the electromagnetic field vector potential in the canonical frame depend only on $u^0$:

$${\mathbb{A}}_a={\alpha }_a\left(u^0\right).$$

(13)

Proof. 

The right side of the system of Equation (10) can be represented in the following form:

$${\left({\xi }_a^{\alpha }{A}_{\alpha }\right)}_{/b}={\left({\xi }_a^{\alpha }{\zeta }_{\alpha }^c{\mathbb{A}}_c\right)}_{/b}={\xi }_a^{\alpha }{\zeta }_{\alpha }^c{\mathbb{A}}_{c/b}.+{\xi }_{a/b}^{\alpha }{\zeta }_{\alpha }^c{\mathbb{A}}_c+{\xi }_a^{\alpha }{\zeta }_{\alpha /b}^c{\mathbb{A}}_c$$

(14)

From (6) it can obtain the equations:

$${\zeta }_{\beta ,\alpha }^a=-{\zeta }_{\gamma }^a{\xi }_{\alpha }^b{\xi }_{b,\beta }^{\gamma }.$$

Using this, one can reduce the expression (14) to the form:

$${\mathbf{A}}_{a/b}={\mathbb{A}}_{c/a}{\xi }_b^{\alpha }{\zeta }_{\alpha }^c+({\xi }_a^{\alpha }{\xi }_{b,\alpha }^{\gamma }-{\xi }_b^{\alpha }{\xi }_{a,\alpha }^{\gamma }){\zeta }_{\gamma }^c{\mathbb{A}}_c={\mathbb{A}}_{c/a}{\xi }_b^{\alpha }{\zeta }_{\alpha }^c+C_{ab}^c{\mathbf{A}}_c.$$

(15)

Thus, from (10), (14), and (15), it follows:

$${\mathbb{A}}_{a/b}=0\Rightarrow {\mathbb{A}}_a={\alpha }_a\left(u^0\right).$$

(16)

□

As it is known (see [81]), the non-holonomic components of the Ricci tensor in the canonical frame depend only on the functions ${\eta }_{\alpha \beta }$ and the structural constants. A similar situation occurs for the energy-momentum tensor of the electromagnetic field, invariant with respect to the group $G_3\left(N\right)$. Let us show that in the non-null homogeneous Petrov space of type $V_4\left(N\right)$ the Maxwell vacuum equations:

$${\displaystyle \frac{1}{G}}{\left(G{F}^{ij}\right)}_{,j}=0$$

(17)

are reduced to a system of algebraic equations on the function ${\eta }_{\alpha \beta }$, which includes the functions ${\alpha }_a$ and also the structure constants. Let us denote the components $F_{ab}$ of the electromagnetic field tensor in the canonical frame ${\zeta }_a^{\alpha }$ as follows:

$$F_{ab}={\zeta }_a^{\alpha }{\zeta }_b^{\beta }{F}_{\alpha \beta },F_{0a}={\zeta }_a^{\alpha }{F}_{0\alpha }={\dot{\alpha }}_a{F}^{ab}={\eta }^{a{a}_1}{\eta }^{b{b}_1}{F}_{a_1{b}_1}$$

From condition (11) it follows:

$$\begin{array}{c}{F}_{ab}=C_{ba}^c{\alpha }_c\Rightarrow F^{ab}=-{\eta }^2(({\eta }^{a2}{\eta }^{b3}-{\eta }^{a3}{\eta }^{b2})C_{23}^c{\alpha }_c+({\eta }^{a3}\eta b1-{\eta }^{a1}{\eta }^{b3})C_{31}^c{\alpha }_c+\\ ({\eta }^{a1}\eta b2-{\eta }^{a2}{\eta }^{b1})C_{12}^c{\alpha }_c.\end{array}$$

Denoting:

$$\begin{array}{c}{f}^1=C_{23}^c{\alpha }_c=C^{1c}{\alpha }_c,f^2=C_{31}^c{\alpha }_c=C^{2c}{\alpha }_c,f^3=C_{12}^c{\alpha }_c=C^{3c}{\alpha }_c,\\ f_1=F^{23},f_2=F^{31},f_3=F^{12},\end{array}$$

one can reduce these relations to the form:

$$f_a={\displaystyle \frac{\epsilon }{{\eta }^2}}{\eta }_{ab}{f}^b.$$

Let us consider the following equations from the system of Maxwell equations:

$${\displaystyle \frac{{\zeta }_{\alpha }^a}{G}}{\left(G{F}^{\alpha i}\right)}_{,i}={\displaystyle \frac{{\zeta }_{\alpha }^a}{\zeta }}{\left(\zeta {\zeta }_{a_1}^{\alpha }{\zeta }_b^{\beta }{F}^{a_{1}b}\right)}_{,\beta }+{\displaystyle \frac{1}{\eta }}{\left(\eta F^{a0}\right)}_{,0}=F^{ab}{C}_{cb}^c+{\displaystyle \frac{1}{2}}{F}^{cb}{C}_{bc}^a+{\displaystyle \frac{1}{\eta }}{\left(\eta {\eta }^{ab}{\dot{\alpha }}_b\right)}_{,0}.$$

(18)

Here we use the expression for the first term on the right-hand side.

$${\displaystyle \frac{{\zeta }_{\alpha }^a}{\zeta }}{\left(\zeta {\zeta }_{a_1}^{\alpha }{\zeta }_b^{\beta }{F}^{a_{1}b}\right)}_{,\beta }=F^{ab}({\zeta }_{b,\alpha }^{\alpha }-{\zeta }_{c|b}^{\alpha }{\zeta }_{\alpha }^c)+F^{cb}{\zeta }_{c,|b}^{\alpha }{\zeta }_{\alpha }^a=F^{ab}{C}_{cb}^c+{\displaystyle \frac{1}{2}}{F}^{cb}{C}_{bc}^a$$

In order to reduce Maxwell equations to a system of ordinary differential equations of the first order, we introduce new independent functions:

$${\beta }^a=\eta {\eta }^{ab}{\dot{\alpha }}_b.$$

In this case, the system of Maxwell equations must be supplemented with the system of equations:

$${\dot{\alpha }}_b={\displaystyle \frac{1}{\eta }}{\eta }_{ab}{\beta }^a.$$

(19)

The system of Equation (18) will take the form:

$$\dot{{\beta }^a}+{\displaystyle \frac{1}{\eta }}{f}^b({\eta }_{1b}(C^{1a}-C_{a3}^a{\delta }_2^a)+{\eta }_{2b}(C^{2a}+C_{a3}^a{\delta }_1^a)+{\eta }_{3b}{C}^{3a})=0.$$

(20)

These Equations (20) should be supplemented by the equation:

$$C_{ab}^a{\beta }^b=0$$

(21)

(it follows from the equations ${\displaystyle \frac{1}{g}}{\left(g{F}^{0\alpha }\right)}_{,\alpha }=0$). Denoting

$$n_{ab}={\displaystyle \frac{1}{\eta }}{\eta }_{ab},Q=C_{a3}^a,$$

one can represent the system of Maxwell equations as the following system of algebraic (with respect to the functions $n_{\alpha \beta }$) equations:

$$n_{ab}{\beta }^b={\dot{\alpha }}_a,$$

(22)

$$f^b(n_{1b}(C^{1a}-Q{\delta }_2^a)+n_{2b}(C^{2a}+Q{\delta }_1^a)+n_{3b}{C}^{3a})=-\dot{{\beta }^a},$$

(23)

$$Q{\beta }^b=0.$$

(24)

For convenience, we represent in the table the values of all quantities included in the Equations (22)–(24). Depending on the number N of the group $G_3\left(N\right)$, they have the following form:

$$\begin{array}{|cccccccccc|}\hline N\hfill & I& II\hfill & III& IV& V& VI& VII& VIII& IX\\ C^{11}\hfill & 0& 1\hfill & 0& 1& 0& 0& -1& 0& 1\\ C^{12}\hfill & 0& 0\hfill & 0& 1& 1& q& q& 0& 0\\ C^{21}\hfill & 0& 0\hfill & -1& -1& -1& -1& 0& 0& 0\\ C^{22}\hfill & 0& 0\hfill & 0& 0& 0& 0& -1& -1& 1\\ C^{33}\hfill & 0& 0\hfill & 0& 0& 0& 0& 0& 0& 1\\ C^{21}+Q\hfill & 0& 0\hfill & 0& 1& 1& q& q& 0& 0\\ C^{12}-Q\hfill & 0& 0\hfill & -1& -1& -1& -1& 0& 0& 0\\ C^{13}\hfill & 0& 0\hfill & 0& 0& 0& 0& 0& 1& 0\\ C^{31}\hfill & 0& 0\hfill & 0& 0& 0& 0& 0& 1& 0\\ f^1\hfill & 0& {\alpha }_1\hfill & 0& {\alpha }_1+{\alpha }_2& {\alpha }_2& q{\alpha }_2& q{\alpha }_2-{\alpha }_1& {\alpha }_3& {\alpha }_1\\ f^2\hfill & 0& 0\hfill & -{\alpha }_1& -{\alpha }_1& -{\alpha }_1& -{\alpha }_1& -{\alpha }_2& -{\alpha }_2& {\alpha }_2\\ f^3\hfill & 0& 0\hfill & 0& 0& 0& 0& 0& {\alpha }_1& {\alpha }_3\\ Q\hfill & 0& 0\hfill & 1& 2& 2& q+1& q& 0& 0\\ \Delta \left(N\right)\hfill & 0& 0\hfill & 0& 1& 1& q& 1& 0& 1\\ \hline\end{array}$$

(25)

Here $\Delta \left(N\right)=C^{11}{C}^{22}+(Q+C^{21})(Q-C^{12}).$

The set of non-null homogeneous Petrov spaces can be divided into three subsets:

I. 

Spaces with solvable groups of motions $G_3\left(I\right)-G_3\left(III\right)({\beta }^3=const);$

II. 

Spaces with solvable groups of motions $G_3\left(IV\right)-G_3\left(VII\right)({\beta }^3=0,Q\Delta \left(N\right)\ne 0);$

III. 

Spaces with non-solvable groups of motions $G_3\left(VIII\right)-G_3\left(IX\right)(Q\Delta \left(N\right)=0).$

In this paper, a unified approach to implementing classification for all types of nonzero homogeneous Petrov spaces is used. The classification is based on the use of solutions of the system of Equation (22) at the initial stage. The set of these solutions (having a common form for all spaces) can be divided into three subsets. Let us represent all these subsets.

Option A ${\beta }^3\ne 0$. The option is not realized for the groups $G_3\left(III\right)-G_3\left(VII\right)$, since for these groups $Q\ne 0\Rightarrow {\beta }^3=0$. It will also not be used as the initial classification step for the group $G_3\left(VIII\right),$ since in this case it is appropriate to start the classification with the option ${\beta }^2\ne 0.$ The solution of the system of Equation (22) has the form:

$$n_{3p}={\displaystyle \frac{1}{{\beta }^3}}({\dot{\alpha }}_p-n_{pq}{\beta }^q),n_{33}={\displaystyle \frac{1}{{{\beta }^3}^2}}({\dot{\alpha }}_3{\beta }^3+{\dot{\alpha }}_p{\beta }^p-n_{pq}{\beta }^q{\beta }^q)(p,q=1,2).$$

(26)

Note that all functions on the right-hand sides of the Equation (26) at this stage are considered as independent functions of the variable $u^0$. After substituting (26) into the system of Equation (23), some of them will remain independent. The other functions will be expressed through these independent functions, as well as through independent functions $\gamma \left(u^0\right),\omega (u^0$), etc., which be introduced to reduce the order of the system of differential Equation (23). This remark is also true for all other types of spaces.

Option B ${\beta }^2\ne 0$. For solvable groups $G_3\left(IV\right)$–$G_3\left(VII\right)$ one has the additional condition ${\beta }^3=0$.

Then the solution of the system of Equation (22) has the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-n_{11}{\beta }^1),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+n_{11}{{\beta }^1}^2),n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{13}{\beta }^1).$$

(27)

Option C ${\beta }^2={\beta }^3=0$. For the groups $G_3\left(V\right)$, $G_3\left(VI\right)$, $G_3\left(VIII\right)$, there exist admissible transformations of canonical vectors that preserve the form of the structure equations. These transformations have the form:

–

for the group $G_3\left(V\right):\widetilde{{\zeta }_1^{\alpha }}={\zeta }_2^{\alpha },\widetilde{{\zeta }_2^{\alpha }}={\zeta }_1^{\alpha }$;

–

for the group $G_3\left(VI\right):\widetilde{{\zeta }_1^{\alpha }}={\zeta }_2^{\alpha },\widetilde{{\zeta }_2^{\alpha }}={\zeta }_1^{\alpha },\widetilde{{\zeta }_3^{\alpha }}=q{\zeta }_3^{\alpha },\tilde{q}={\displaystyle \frac{1}{q}}$;

–

for the group $G_3\left(VIII\right):\widetilde{{\zeta }_1^{\alpha }}={\zeta }_3^{\alpha },\widetilde{{\zeta }_3^{\alpha }}={\zeta }_1^{\alpha }$.

Therefore, for these groups, the option C is a special case of the option B, and it does not considered separately. For the remaining groups, the solution of the system of Equation (22) has the form:

$$n_{1a}={\displaystyle \frac{1}{{\beta }^1}}{\dot{\alpha }}_a.$$

(28)

Option D ${\beta }^a=0.$ From the Equation (22) for all solvable groups groups $G_3\left(N\right)$ it follows:

${\alpha }_a=a_a=const.$ For non-solvable groups, the electromagnetic field is absent.

Holonomic components of the metric tensor and vector potential of the electromagnetic field are given by the Formulas (7) and (12):

$$g_{\alpha \beta }={{\zeta }_{\alpha }}^a{{\zeta }_{\beta }}^b{\eta }^{ab}\left(u^0\right),A_{\alpha }={\zeta }_{\alpha }^a{\mathbb{A}}_a={\zeta }_{\alpha }^a{\alpha }_a.$$

(29)

Since for all homogeneous Petrov spaces the pairs of vector fields ${\xi }_a^{\alpha }$ and ${\zeta }_a^{\alpha }$ have already been found (see [12]), the solutions of the vacuum Maxwell equations in the holonomic coordinate system are determined by the functions $n_{ab},{\alpha }_a$. The types of these functions are determined from the solution of the system of Equations (22)–(24) and are given below for each Petrov space.

The following notations are used below:

$$\varsigma ,{\varsigma }_a,\xi ,{\xi }_a=\pm 1,a,a_a,b,b_a,c,c_a=const.$$

4. Solutions of Maxwell Equations for Groups ${\mathbf{G}}_{\mathbf{3}}(\mathbf{N}<\mathbf{IV})$

4.1. Group $G_3$(I)

$G_3\left(I\right)$ is an Abelian group, and all structure constants are zero $\Rightarrow {\xi }_a^{\alpha }={\zeta }_a^{\alpha }={\delta }_a^{\alpha }$. From Maxwell equations, it follows:

$${\beta }^a=const.$$

Vector ${\beta }^a$ can be diagonalized by admissible coordinate transformations:

$${\beta }^a={\displaystyle \frac{1}{c}}{\delta }_3^a.$$

The solution can be written as

$${\eta }_{ab}={\displaystyle \frac{\epsilon n_{ab}}{det∥n_{a_1{b}_1}∥}},n_{a3}=c{\dot{\alpha }}_a.$$

${\alpha }_a$ and other components $n_{ab}$ are arbitrary functions. Note that in the case of the Abelian group, holonomic and non-holonomic coordinate systems coincide.

4.2. Group $G_3$(II)

The Killing vector fields ${\xi }_a^{\alpha }$ and the vector fields of the canonical frame ${\zeta }_a^{\alpha }$ (as it has been already noted, these vectors can be swapped) have the form:

$${\xi }_a^{\alpha }={\delta }_a^1{\delta }_1^{\alpha }+{\delta }_a^2{\delta }_2^{\alpha }+{\delta }_a^3({\delta }_3^{\alpha }+u^2{\delta }_1^{\alpha }).$$

(30)

$${\zeta }_a^{\alpha }={\delta }_1^{\alpha }\left({\delta }_a^1+u^3{\delta }_a^3\right)+{\delta }_2^{\alpha }{\delta }_a^2+{\delta }_3^{\alpha }{\delta }_a^3.$$

The Equation (23) are reduced to the following:

$${\dot{\beta }}^a=-{\delta }_1^a{\alpha }_1{n}_{11}\Rightarrow {\dot{\beta }}^1=-{\alpha }_1{n}_{11},{\beta }^2=c_2,{\beta }^3=c_3(c_a=const).$$

(31)

Let us consider all possible options.

Option $\mathbf{A}.{\beta }^3=1.$ Using admissible transformations of vectors ${\zeta }_p^{\alpha }(p,q=2,3),$ one of the parameters $c_p$ (for example, $c_2={\beta }^2$) can be set to zero. Then the Equation (22) takes the form:

$$n_{13}={\dot{\alpha }}_1-n_{11}{\beta }^1,n_{23}={\dot{\alpha }}_2-n_{12}{\beta }^1,n_{33}={\dot{\alpha }}_3-{\dot{\alpha }}_1{\beta }^1+n_{11}{{\beta }^1}^2.$$

The Equation (31), has next solutions:

Variant ${\mathbf{A}}_{\mathbf{1}}{\alpha }_1\ne 0$,

$$n_{11}={\displaystyle \frac{1}{{\alpha }_1}}{\dot{\beta }}^1,n_{13}={\displaystyle \frac{{\dot{\alpha }}_1{\alpha }_1+{\beta }^1{\dot{\beta }}_1}{{\alpha }_1}},n_{23}={\dot{\alpha }}_2-n_{12}{\beta }^1,n_{33}={\dot{\alpha }}_3-{\beta }^1{\displaystyle \frac{{\dot{\alpha }}_1{\alpha }_1+{\beta }^1{\dot{\beta }}_1}{{\alpha }_1}}.$$

Variant ${\mathbf{A}}_{\mathbf{2}}{\alpha }^1=0,{\beta }^1=c,$

$$n_{13}=-c{n}_{11},n_{23}={\dot{\alpha }}_2-c{n}_{12},n_{33}={\dot{\alpha }}_3+c^2{n}_{11},$$

$n_{11}$ is an arbitrary function.

Option $\mathbf{B}{\beta }^3=0,{\beta }^2=1.$ It is equivalent to the previous option.

Option $\mathbf{C}{\beta }^3={\beta }^2=0,{\beta }^1\ne 0,n_{1a}={\displaystyle \frac{1}{{\beta }^1}}\dot{{\alpha }_a}.$ Solution of the Equation (31) have the form:

$${\alpha }_1=asin\omega ,{\beta }^1=acos\omega ,n_{11}=\dot{\omega },n_{12}={\displaystyle \frac{\dot{{\alpha }_2}}{acos\omega }},n_{13}={\displaystyle \frac{\dot{{\alpha }_3}}{acos\omega }}.$$

4.3. Group $G_3$(III)

The Killing vector fields ${\xi }_a^{\alpha }$ and the canonical frame ${\zeta }_a^{\alpha }$ can be represented in the form:

$${\xi }_a^{\alpha }={\delta }_a^1{\delta }_1^{\alpha }+{\delta }_a^2{\delta }_2^{\alpha }+{\delta }_a^3({\delta }_3^{\alpha }+u^2{\delta }_2^{\alpha }),$$

(32)

$${\zeta }_{\left(a\right)}^{\alpha }={\delta }_1^{\alpha }{\delta }_a^1+{\delta }_2^{\alpha }{\delta }_a^{2}exp\left(-u_3\right)+{\delta }_3^{\alpha }{\delta }_a^3.$$

Equation (23) have the form:

$${\dot{\beta }}^1=0\Rightarrow {\beta }^1=c=const,{\dot{\beta }}^2=-{\alpha }_1{n}_{12}.$$

(33)

Since in this case from Equation (33), it follows ${\beta }^3=0$, option $\mathbf{A}$ does not need to be realized. Therefore, only options $\mathbf{B},\mathbf{C},\mathbf{D}$ should be considered.

Option $\mathbf{B}{\beta }^2\ne 0,{\beta }^1=c.$

Equation (33) need to be supplemented with Equation (22), having in this case the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-c{n}_{11}),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_{1}c+c^2{n}_{11}),n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{13}c).$$

(34)

From the relations (33), (34) it follows

$${({{\alpha }_1}^2+{{\beta }^2}^2)}_{,0}=2c{\alpha }_1{n}_{11}.$$

Using this relation, one finds all the corresponding solutions of Maxwell equations.

Variant ${\mathbf{B}}_{\mathbf{1}}$ ${\alpha }_1\ne 0\Rightarrow n_{21}=-{\displaystyle \frac{\dot{{\beta }^2}}{{\alpha }_1}},c{n}_{11}={\displaystyle \frac{\dot{{\alpha }_1}{\alpha }_1+{\beta }^2\dot{{\beta }^2}}{{\alpha }_1}};$

1

${\beta }^1=c=0,{\alpha }_1=asin\omega ,{\beta }^2=acos\omega ,n_{12}=\dot{\omega },n_{11}$ is an arbitrary function.

2

${\beta }^1=c,{\beta }^1=c\Rightarrow {\beta }^2=1.n_{11}={\displaystyle \frac{\dot{{\alpha }_1}{\alpha }_1+{\beta }^2\dot{{\beta }^2}}{c{\alpha }_1}}{n}_{12}=\dot{\omega },n_{12}$ is an arbitrary function.

Variant ${\mathbf{B}}_{\mathbf{2}}{\alpha }_1=0\Rightarrow {\beta }^2=1,n_{12}=-c{n}_{11}.n_{11}$ is an arbitrary function.

Option $\mathbf{C}$ ${\beta }^2=0$. From Equations (22) and (28) it follows: ${\beta }^1=1,{\alpha }_1{n}_{12}=0,n_{1a}=\dot{{\alpha }_a}$. The solution has the form:

1

${\alpha }_2=a,n_{12}=0,$

2

${\alpha }_1=0,n_{11}=0.n_{12}$ is an arbitrary function.

Option $\mathbf{D}$ ${\alpha }_a=const,{\beta }^a=0$. From Equation (35) there follow two solutions of Equation (23):

1

${\alpha }_1=0,$

2

$n_{12}=0.$

5. Solutions of Maxwell Equations for the Groups ${\mathbf{G}}_{\mathbf{3}}$(III < N < VIII)

From the system of Equation (23), it follows:

$$n_{11}{f}^1+n_{12}{f}^2={\displaystyle \frac{\dot{{\beta }^2}(Q+C^{21})-\dot{{\beta }^1}{C}^{22}}{\Delta \left(N\right)}},n_{12}{f}^1+n_{22}{f}^2={\displaystyle \frac{\dot{{\beta }^1}(C^{12}-Q)-\dot{{\beta }^2}{C}^{11}}{\Delta \left(N\right)}}.$$

(35)

Since Option $\mathbf{A}$ does not be realized (${\beta }^3=0$), only Option $\mathbf{B},\mathbf{C},\mathbf{D}$ should be considered.

Option B. ${\beta }^2\ne 0,{\beta }^3=0$,

From Equation (22) we find additional conditions on functions $n_{2a}:$

$$n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+n_{11}{{\beta }^1}^2),n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-n_{11}{\beta }^1),n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{12}{\beta }^1).$$

Using these conditions in the system of Equation (35), we reduce this system to the form:

$$n_{11}({\beta }^2{f}^1-{\beta }^1{f}^2)={\displaystyle \frac{{\beta }^2}{\Delta }}(C^{22}{\dot{\beta }}^1-(C^{21}+Q){\dot{\beta }}^2)-f^2{\dot{\alpha }}_1,$$

(36)

$$\Delta \left(N\right)({\dot{\alpha }}_1{f}^1+{\dot{\alpha }}_2{f}^2)=(Q-C^{12}){\dot{\beta }}^1{\beta }^2+C^{11}{\dot{\beta }}^2{\beta }^2+C^{22}{\dot{\beta }}^1{\beta }^1-(C^{21}+Q){\dot{\beta }}^2{\beta }^1.$$

(37)

Thus, the solutions of Maxwell equations for this Opyion $\mathbf{B}$ fall into two types:

Variant ${\mathbf{B}}_{\mathbf{1}}({\beta }^2{f}^1-{\beta }^1{f}^2)\ne 0\Rightarrow$

$$n_{11}={\displaystyle \frac{{\beta }_2}{\Delta \left(N\right)({\beta }^2{f}^1-{\beta }^1{f}^2)}}(C^{22}{\dot{\beta }}^1-(C^{21}+Q){\dot{\beta }}^2)-f^2{\dot{\alpha }}_1).$$

(38)

The functions ${\alpha }_p,{\beta }^p$ are related by the condition (37).

Variant ${\mathbf{B}}_{\mathbf{2}}({\beta }^2{f}^1-{\beta }^1{f}^2)=0$,

$n_{11}$ is an arbitrary function. In addition to the condition (37), two conditions more are imposed on the functions ${\alpha }_p,{\beta }^p$:

$${\beta }^2{f}^1={\beta }^1{f}^2,{\beta }^2(C^{22}{\dot{\beta }}^1-(C^{21}+Q){\dot{\beta }}^2)=f^2{\dot{\alpha }}_1\Delta \left(N\right).$$

(39)

Option C ${\beta }^2={\beta }^3=0.$

From the conditions (23) it follows $n_{11}={\displaystyle \frac{{\dot{\alpha }}_1}{{\beta }^1}},n_{12}={\displaystyle \frac{{\dot{\alpha }}_2}{{\beta }^1}},n_{13}={\displaystyle \frac{{\dot{\alpha }}_3}{{\beta }^1}}$ Substitute these expressions into (35). As a result, we obtain:

$$n_{22}{f}^2={\displaystyle \frac{1}{\Delta }}(Q-C^{12}){\dot{\beta }}^1-f^1{\dot{\alpha }}_2,$$

(40)

$${\displaystyle \frac{1}{\Delta }}{C}^{22}{\dot{\beta }}^1=f^1{\dot{\alpha }}_1+f^2{\dot{\alpha }}_2.$$

(41)

Thus, for this, two variants of solutions are possible.

Variant ${\mathbf{C}}_{\mathbf{1}}{f}^2\ne 0$.

$$n_{22}={\displaystyle \frac{1}{f^2}}({\displaystyle \frac{1}{\Delta }}(Q-C^{12}){\dot{\beta }}^1-f^1{\dot{\alpha }}_2),$$

(42)

The functions ${\alpha }_p,{\beta }^1$ are related by the condition (41).

Variant ${\mathbf{C}}_{\mathbf{2}}{f}^2=0\Rightarrow n_{22}-$ arbitrary function of $u^0$.

In addition to the condition (41), another condition is imposed on the functions ${\alpha }_p,{\beta }^1$, which follows from the Equation (40):

$${\displaystyle \frac{1}{\Delta }}(Q-C^{12}){\dot{\beta }}^1=f^1{\dot{\alpha }}_2.$$

(43)

Option D ${\beta }^a=0\Rightarrow {\alpha }_a=const.$

In this case, the system (23) can be represented as

$$n_{11}{f}^1+n_{12}{f}^2=0,$$

(44)

$$n_{12}{f}^1+n_{22}{f}^2=0.$$

System (44) must have nonzero solutions, otherwise the electromagnetic field is absent. Therefore

$$det∥n_{pq}∥\Rightarrow n_{pq}={\gamma }_p{\gamma }_q\Rightarrow {\gamma }_p{f}^p=0({\gamma }_p={\gamma }_p\left(u^0\right)).$$

5.1. Group $G_3\left(IV\right)$

Let us present the Killing vector fields:

$${\xi }_a^{\alpha }={\delta }_1^{\alpha }{\delta }_a^{1}exp(-u^3)+{\delta }_2^{\alpha }{\delta }_a^2+{\delta }_a^3({\delta }_3^{\alpha }+u^2({\delta }_2^{\alpha }+{\delta }_1^{\alpha }exp(-u^3)).$$

(45)

and the vector fields of the canonical frame:

$${\zeta }_a^{\alpha }={\delta }_1^{\alpha }({\delta }_a^1+u^2{\delta }_a^3)exp(-u^3)+{\delta }_2^{\alpha }({\delta }_a^2+u^2{\delta }_a^3)+{\delta }_a^3{\delta }_3^{\alpha }.$$

The system of Equation (36) has the form:

$$n_{12}({\alpha }_1+{\alpha }_2)-n_{22}{\alpha }_1=-({\dot{\beta }}^1+{\dot{\beta }}^2),$$

(46)

$$n_{11}({\alpha }_1+{\alpha }_2)-n_{12}{\alpha }_1={\dot{\beta }}^2.$$

Consider the

Option $\mathbf{B}$.

Components $n_{p2}(p,q=1,2)$ have the form (see (22)):

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-n_{11}{\beta }^1),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+n_{11}{{\beta }^1}^2).$$

Using the conditions (22) and (37), one can represent Equation (46) in the form:

$$n_{11}({\beta }^2({\alpha }_1+{\alpha }_2)+{\beta }^1{\alpha }_1)={\beta }^2{\dot{\beta }}^2+{\alpha }_1{\dot{\alpha }}_1,$$

(47)

$${\dot{\alpha }}_1({\alpha }_1+{\alpha }_2)-{\dot{\alpha }}_2{\alpha }_1+({\dot{\beta }}^1+{\dot{\beta }}^2){\beta }^2-{\dot{\beta }}^2{\beta }^1=0.$$

(48)

Let us consider all possible variants.

Variant ${\mathbf{B}}_{\mathbf{1}}{\beta }^2({\alpha }_1+{\alpha }_2)+{\beta }^1{\alpha }_1\ne 0.$ In this case:

$$n_{11}={\displaystyle \frac{1}{({\beta }^2({\alpha }_1+{\alpha }_2)+{\beta }^1{\alpha }_1)}}({\beta }^2{\dot{\beta }}^2+{\alpha }_1{\dot{\alpha }}_1).$$

(49)

If the function ${\alpha }_1=0,$ one can reduced Equation (47) to the form:

$${(ln{\beta }^2+{\displaystyle \frac{{\beta }_1}{{\beta }^2}})}_{,0}=0.$$

(50)

Hence, the first solution is:

1

${\alpha }_1=0,{\beta }_1={\beta }_2(c-ln{\beta }_2).$

Consider the case when ${\alpha }_1\ne 0.$ Let us introduce new independent functions as follows:

$${\alpha }_1=exp\sigma cos\omega ,{\beta }^2=exp\sigma sin\omega ,{\alpha }_2={\gamma }_{2}cos\omega ,{\beta }^1={\gamma }_{1}sin\omega .$$

The Equation (48) will take the form:

$${cos}^2\left(\omega \right)(\dot{{\gamma }_1}+\dot{{\gamma }_2})=\dot{\sigma }+\dot{{\gamma }_1}\Rightarrow {sin}^2\left(\omega \right)(\dot{{\gamma }_1}+\dot{{\gamma }_2})=\dot{{\gamma }_2}-\dot{\sigma }.$$

(51)

The Equation (51) has two independent solutions:

2

${\alpha }_1=exp\sigma cos\omega ,{\beta }^2=exp\sigma sin\omega ,{\alpha }_2=\sigma exp\sigma cos\omega ,{\beta }^1=-\sigma exp\sigma sin\omega .$

3

$cos\left(\omega \right)=\sqrt{{\displaystyle \frac{\dot{\sigma }+\dot{{\gamma }_1}}{\dot{{\gamma }_1}+\dot{{\gamma }_2}}}},sin\left(\omega \right)=\sqrt{{\displaystyle \frac{\dot{{\gamma }_2}-\dot{\sigma }}{\dot{{\gamma }_2}+\dot{{\gamma }_1}}}}\Rightarrow$

$${\alpha }_1=exp\sigma \sqrt{{\displaystyle \frac{\dot{\sigma }+\dot{{\gamma }_1}}{\dot{{\gamma }_1}+\dot{{\gamma }_2}}}},{\beta }^2=exp\sigma \sqrt{{\displaystyle \frac{\dot{{\gamma }_2}-\dot{\sigma }}{\dot{{\gamma }_2}+\dot{{\gamma }_1}}}},,$$

$${\alpha }_2={\gamma }_{2}exp\sigma \sqrt{{\displaystyle \frac{\dot{\sigma }+\dot{{\gamma }_1}}{\dot{{\gamma }_1}+\dot{{\gamma }_2}}}},{\beta }^1={\gamma }_{1}exp\sigma \sqrt{{\displaystyle \frac{\dot{{\gamma }_2}-\dot{\sigma }}{\dot{{\gamma }_2}+\dot{{\gamma }_1}}}}$$

Variant ${\mathbf{B}}_{\mathbf{2}}{n}_{11}$ is an arbitrary function, ${\beta }^2{\dot{\beta }}^2-{\alpha }_1{\dot{\alpha }}_1=0\Rightarrow {\alpha }_1=pcos\omega ,{\beta }^2=psin\omega$. The functions ${\alpha }_2,{\beta }^1$ obey the equations:

$${{\alpha }_1}^2{({\displaystyle \frac{{\dot{\alpha }}_2}{{\alpha }_1}})}_{,0}={{\beta }^2}^2{({\displaystyle \frac{{\dot{\beta }}^2}{{\beta }^2}})}_{,0},{\beta }^2({\alpha }_1+{\alpha }_2)+{\beta }^1{\alpha }_1=0.$$

(52)

The solution of the system of Equation (52) can be represented as the following:

4

${\alpha }_1=(c_1+c_2)cos\omega ,{\beta }^2=(c_1+c_2)sin\omega ,{\alpha }_2=-c_{2}cos\omega ,{\beta }^1=-c_{1}sin\omega .$

Let us consider the

Option C ${\beta }^1\ne 0,{\beta }^2=0.n_{1a}={\displaystyle \frac{{\alpha }_a}{{\beta }^1}}.$ Maxwell equations takes the form:

$${\dot{\alpha }}_1({\alpha }_1+{\alpha }_2)-{\dot{\alpha }}_2{\alpha }_1=0,n_{22}{\beta }^1{\alpha }_1=({\dot{\alpha }}_2{\alpha }_2+{\dot{\beta }}^1{\beta }^1)+\dot{{\alpha }_2}{\alpha }_1.$$

(53)

There are two solutions of the Equation (53):

5

${\alpha }_1=0,n_{11}=0,n_{12}=\dot{\omega }.{\alpha }_2=acos\omega ,{\beta }^1=asin\omega .$$n_{22}$ is an arbitrary function.

6

$n_{22}={\displaystyle \frac{1}{{\beta }^1{\alpha }_1}}({\dot{\alpha }}_2{\alpha }_2+{\dot{\beta }}^1{\beta }^1+\dot{{\alpha }_2}{\alpha }_1).{\alpha }_2={\alpha }_1(c+ln{\alpha }_1),n_{11}$ is an arbitrary function.

Let us consider the:

Option D ${\beta }^a=0,{\alpha }_a=a_a=const\Rightarrow$ Equation (44) have the form:

$$n_{12}(a_1+a_2)-n_{22}{a}_1-0,n_{11}(a_1+a_2)-n_{12}{a}_1-0.$$

There are two independent solutions:

1

$a_1=0,n_{11}=n_{12}=0;$

2

$a_2=c{a}_1,n_{12}=n_{11}(1+c),n_{22}=n_{11}{(1+c)}^2.$

5.2. Groups $G_3\left(V\right)and{G}_3\left(VI\right)$

Maxwell equations for the group $G_3\left(V\right)$ are obtained from Maxwell equations for the group $G_3\left(VI\right)$ with the zero value of the parameter q. Therefore, all solutions for the group $G_3\left(V\right)$ are special cases of solutions for the group $G_3\left(VI\right)$, and we will not consider the group $G_3\left(V\right)$ separately.

Using structural equations and Equation (6), one can obtain the following pair of sets of vectors: ${\xi }_a^{\alpha },$ and ${\zeta }_a^{\alpha }$:

$${\zeta }_a^{\alpha }={\delta }_1^{\alpha }{\delta }_a^1+{\delta }_2^{\alpha }{\delta }_a^{2}exp\left(q{u}^3\right)+({\delta }_3^{\alpha }-{\delta }_a^{2}q{u}^1){\delta }_a^3,{\xi }_a^{\alpha }={\delta }_1^{\alpha }{\delta }_a^{1}exp(-u^3)+{\delta }_2^{\alpha }{\delta }_a^2+({\delta }_3^{\alpha }+{\delta }_a^{2}q{u}^2){\delta }_a^3.$$

(54)

The system of Equation (23) has the form:

$$q{n}_{11}{\alpha }_2-n_{12}{\alpha }_1={\dot{\beta }}^2,$$

(55)

$$q^2{n}_{12}{\alpha }_2-q{\alpha }_1{n}_{22}=-{\dot{\beta }}^1.$$

Let us consider the

Option B ${\beta }^2\ne 0,{\beta }^3=0,n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-n_{11}{\beta }^1),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+n_{11}{{\beta }^1}^2).$

The Equation (55) will take the form:

$$n_{11}({\beta }^1{\alpha }_1+q{\beta }^2{\alpha }_2)={\alpha }_1{\dot{\alpha }}_1+{\beta }^2{\dot{\beta }}^2,$$

(56)

$$q(q{\dot{\alpha }}_1{\alpha }_2-{\dot{\alpha }}_2{\alpha }_1)+{\dot{\beta }}^1{\beta }^2-q{\dot{\beta }}^2{\beta }^1=0.$$

(57)

Let us find all solutions of these equations.

Variant ${\mathbf{B}}_{\mathbf{1}}$ Let ${\beta }^1{\alpha }_1+q{\beta }^2{\alpha }_2\ne 0.$ Then:

$$n_{11}={\displaystyle \frac{1}{{\beta }^1{\alpha }_1+q{\beta }^2{\alpha }_2}}({\alpha }_1{\dot{\alpha }}_1+{\beta }^2{\dot{\beta }}^2),$$

(58)

There are the following independent solutions of the Equation (57).

• ${\alpha }_1=0,{\beta }^1=c{{\beta }^2}^q.$
• ${\alpha }_2=c_2{{\alpha }_1}^q,{\beta }^1=c_1{{\beta }^2}^q.$
• ${\alpha }_1={\beta }^2{\left({\displaystyle \frac{\dot{\gamma }}{\dot{\omega }}}\right)}^{{\displaystyle \frac{1}{q+1}}},{\alpha }_2=\omega {\left({\beta }^2\right)}^q\left({\displaystyle \frac{\dot{\gamma }}{\dot{\omega }}}\right),{\beta }_1=q\gamma {\left({\beta }^2\right)}^q.$

Variant ${\mathbf{B}}_{\mathbf{2}}$ Two more additional equations for the functions ${\alpha }_p,{\beta }^p$ appear:

$${\beta }^1{\alpha }_1+q{\beta }^2{\alpha }_2=0,{\beta }^2{\dot{\beta }}^2+{\alpha }_1{\dot{\alpha }}_1=0.$$

Then the solutions of Equation (57) have the following form:

$${\beta }^1=-qbsin\omega ,{\alpha }_2=bcos\omega ,{\alpha }_1=acos\omega ,{\beta }^2=asin\omega .$$

Option C Due to the existing symmetry of Maxwell equations with respect to the permutation $1\iff 2,$ variant C is a special case of the variant B. That is why we will not consider it separately.

Option D ${\beta }^a=0,{\alpha }_a=b_a=const$. Due to the symmetry mentioned above, there is a unique independent solution of the Equations (32) and (74):

$$a_2=0,n_{12}=n_{22}=0.$$

5.3. Group $G_3\left(VII\right)$

Let us choose vector fields of the canonical frame in the form:

$${\zeta }_a^{\alpha }={\delta }_a^1({\delta }_1^{\alpha }+{\delta }_3^{\alpha }(u^2+2u^{3}cosa)))+{\delta }_{\alpha }^2{\delta }_a^2+{\delta }_{\alpha }^3{\delta }_a^3.$$

(59)

Then from the Equation (6), it follows:

$${\xi }_a^{\alpha }={\delta }_1^{\alpha }{\delta }_a^1+exp\gamma ({\delta }_2^{\alpha }({\delta }_a^{2}sin\sigma -{\delta }_a^{3}cos\sigma )+{\delta }_3^{\alpha }({\delta }_a^{2}sin(p+a)-{\delta }_a^{3}cos(\sigma +a))).$$

(60)

Here $\sigma =u^{0}sina,\gamma =u^{0}cosa$.

The system of Equation (23) takes the form:

$$n_{11}(q{\alpha }_2-{\alpha }_1)-n_{12}{\alpha }_2={\dot{\beta }}^1+q{\dot{\beta }}^2,$$

(61)

$$n_{12}(q{\alpha }_2-{\alpha }_1)-n_{22}{\alpha }_2={\dot{\beta }}^2.$$

Consider

Option B

Using Equation (27), one can represent Equation (61) in the form:

$$n_{11}({\beta }^1{\alpha }_2+{\beta }^2(q{\alpha }_2-{\alpha }_1))={\alpha }_2{\dot{\alpha }}_1+{\beta }^2({\dot{\beta }}^1+q{\dot{\beta }}^2),$$

(62)

$${\dot{\alpha }}_1{\alpha }_1+{\dot{\alpha }}_2{\alpha }_2+{\beta }^1{\dot{\beta }}^1+{\beta }^2{\dot{\beta }}^2=q({\dot{\alpha }}_1{\alpha }_2-{\beta }^1{\dot{\beta }}^2).$$

(63)

Let us consider this system.

${\mathbf{B}}_{\mathbf{1}}{\beta }^1{\alpha }_2+{\beta }^2(q{\alpha }_2-{\alpha }_1)\ne 0.$ From Equation (62), it follows:

$$n_{11}={\displaystyle \frac{1}{({\beta }^1{\alpha }_2+{\beta }^2(q{\alpha }_2-{\alpha }_1))}}({\alpha }_2{\dot{\alpha }}_1+{\beta }^2({\dot{\beta }}^1+q{\dot{\beta }}^2)).$$

(64)

To solve the Equation (63), let us introduce the function $\gamma$:

$$2q\gamma ={{\alpha }_1}^2+{{\alpha }_2}^2+{{\beta }_1}^2+{{\beta }_2}^2.$$

(65)

Then Equation (63) takes the form:

$$\dot{\gamma }=({\dot{\alpha }}_1{\alpha }_2-{\dot{\beta }}_2{\beta }_1).$$

(66)

The derivatives of the functions ${\alpha }_2,{\beta }^1$ are not presented in the Equation (63). By eliminating one of them (${\alpha }_2$ or ${\beta }^1$) from the system of Equation (65) one obtains an equation for the remaining function. These functions will be expressed through the functions ${\alpha }_1,{\beta }^2,\gamma$ and their first derivatives. Let us consider all possible options.

• $\dot{\gamma }=0\Rightarrow {\dot{\alpha }}_1={\dot{\beta }}^2=0\Rightarrow {\alpha }_1=a,{\beta }^2=b,{\alpha }_2=ccos\omega ,{\beta }^1=csin\omega .$
• ${\dot{\alpha }}_1\ne 0.$ From Equations (65) and (66) it follows:

$${\alpha }_2={\displaystyle \frac{\varsigma }{\dot{{\alpha }_1}}}\sqrt{\dot{\gamma }+{\beta }_1\dot{{\beta }^2}},{\beta }_1={\displaystyle \frac{1}{{\left(\dot{{\alpha }_1}\right)}^2+{\left(\dot{{\beta }^2}\right)}^2}}(\dot{\gamma }\dot{{\beta }^2}+\Phi ),$$

(67)

$${\Phi }^2=({\left(\dot{\gamma }\dot{{\beta }^2}\right)}^2+({\dot{{\alpha }_1}}^2+{\dot{{\beta }^2}}^2)({\dot{{\alpha }_1}}^2(2q\gamma -{{\alpha }_1}^2-{{\beta }^2}^2)-{\dot{\gamma }}^2).$$

Let us consider the

Variant ${\mathbf{B}}_{\mathbf{2}}$ ${\beta }^1{\alpha }_2+{\beta }^2(q{\alpha }_2-{\alpha }_1)=0\Rightarrow n_{11}$—arbitrary function of the variable $u^0$.

The functions ${\alpha }_a,{\beta }^a$ obey the Equation (63) and the system of equations:

$${\alpha }_2({\beta }^1+q{\beta }^2)={\alpha }_1{\beta }^2,{\beta }^2({\dot{\beta }}^1+q{\dot{\beta }}^2)={\alpha }_2{\dot{\alpha }}_1.$$

(68)

From Equation (68), it follows:

$${\alpha }_2(({\beta }^1+q{\beta }^2)(\dot{{\beta }^1}+q\dot{{\beta }^2})+{\alpha }_1\dot{{\alpha }_1})\Rightarrow ({\beta }^1+q{\beta }^2)(\dot{{\beta }^1}+q\dot{{\beta }^2})+{\alpha }_1\dot{{\alpha }_1}=0.$$

(69)

Therefore, the solutions of the Equations (63), (68) and (69) have the form:

1

${\alpha }_1=asin\omega ,{\alpha }_2=bsin\omega ,{\beta }_1=a(p-qb)cos\omega ,{\beta }_2=bcos\omega ,$

2

${\alpha }_1={\alpha }_2=0,{\beta }^1=const,{\beta }^2=const.$

Let us consider the

Option $\mathbf{C}{\beta }^2={\beta }_3=0.$ Equation (61) have the form:

$$n_{11}({\alpha }_1-q{\alpha }_2)+n_{12}a{\alpha }_2=-\dot{{\beta }^1},n_{12}({\alpha }_1-q{\alpha }_2)+n_{22}{\alpha }_2=0.$$

(70)

Using the system of Equation (28):

$$n_{1a}={\displaystyle \frac{\dot{{\alpha }_a}}{{\beta }^1}},$$

one can represents the system of Equation (61) in the following form:

$$n_{22}{\beta }^1{\alpha }_2={\dot{\alpha }}_2(q{\alpha }_2-{\alpha }_1),$$

(71)

$${\dot{\alpha }}_1{\alpha }_1+{\dot{\alpha }}_2{\alpha }_2+{\beta }^1{\dot{\beta }}^1=q{\dot{\alpha }}_1{\alpha }_2.$$

(72)

The Equations (71) and (72) have the next nonequivalent solutions:

1

${\alpha }_2\ne 0n_{22}=(q{\alpha }_2-{\alpha }_1){\displaystyle \frac{\dot{{\alpha }_2}}{{\beta }_1{\alpha }_2}},$

(a)

${\alpha }_2={\displaystyle \frac{\dot{\gamma }}{\dot{{\alpha }_1}}},{\beta }_1={\displaystyle \frac{\varsigma }{\dot{{\alpha }_1}}}\sqrt{{\dot{{\alpha }_1}}^2(2q\gamma -{{\alpha }_1}^2)-{\dot{\gamma }}^2},$

(b)

${\alpha }_1=a,{\alpha }_2=pcos\omega ,{\beta }^1=psin\omega .$

2

${\alpha }_2=0\Rightarrow n_{22}$ is an arbitrary function, ${\alpha }_1=acos\omega ,{\beta }^1=asin\omega .$

Let us consider the

Option $\mathbf{D}{\beta }^a=0,{\alpha }_b=a_b=const$. Equation (61) take the form:

$$n_{11}(q{a}_2-a_1)-n_{12}{a}_2=0,n_{12}(q{a}_2-a_1)-n_{22}{a}_2=0.$$

Since in the case of the group $G_3\left(VII\right)$, the indices 1, 2 enter into Maxwell equations and into the components of the metric tensor $g_{ij}$ non-symmetrically, there are two nonequivalent solutions:

1

$a_1=a_{2}q,n_{12}=n_{22}=0$,

2

$a_2=0,n_{12}=n_{11}=0$.

6. Solutions of Maxwell Equations for Unsolvable Groups

6.1. Group $G_3$(VIII)

The set of generators of the group $G_3\left(VIII\right)$ for homogeneous Petrov spaces can be represented as

$$X_1=p_{3}exp\left(-u^2\right),X_2=p_2,X_3=\left(p_1-u^3{p}_2+{\displaystyle \frac{1}{2}}\left({u^3}^2-ϵ\right)p_3\right)exp{u}^2(ϵ=0,\pm 1).$$

(73)

In the book [10], the set of vectors (73) (more precisely: equivalent to the set (73)) was used to classify homogeneous Petrov null spaces. At the same time, for solving the classification problem for homogeneous Petrov non-null spaces, it has been used with the set of vectors (74):

$${\mathrm{X}}_1=p_{3}exp\left(-u^2\right),{\mathrm{X}}_2=p_2,{\mathrm{X}}_3=(p_1-u^3{p}_2+{\displaystyle \frac{1}{2}}{u^3}^2{p}_3)exp{u}^2,$$

(74)

(see [17], p. 157). In the papers [12,80] the nonequivalence of the sets (73), (74) has been proven. Therefore, the classification of homogeneous non-null Petrov spaces with the group of motions $G_3\left(VIII\right)$ should be supplemented. The easiest way to do this is to assume that the group operators (73) are defined by the vectors of the canonical frame ${\zeta }_a^{\alpha }$. Then:

$$g^{\alpha \beta }={\zeta }_a^{\alpha }{\zeta }_b^{\beta }{\eta }^{ab},Y_a={\zeta }_a^{\alpha }{p}_a,$$

where

$${\zeta }_a^{\alpha }={\delta }_a^1{\delta }_3^{\alpha }exp(-u^2)+{\delta }_a^2{\delta }_2^{\alpha }+{\delta }_a^3({\delta }_1^{\alpha }-u^3{\delta }_2^{\alpha }+{\displaystyle \frac{1}{2}}({u^3}^2-ϵ){\delta }_3^{\alpha })exp{u}^2.$$

(75)

As already noted, the classification results using the canonical frame differ from the classification results using Petrov’s method only in the choice of the holonomic coordinate system. Obviously, these results do not depend on the choice of values of the parameter $ϵ$ (the choice of $ϵ$ affects the form of the holonomic components of the metric tensor ($g_{\alpha \beta }$) and the holonomic components of the electromagnetic field vector $(A_{\alpha }$)).

The structure constants for (75) are of the form:

$$C_{12}^1=C^{31}=1,C_{23}^1=C^{13}=1,C_{31}^2=C^{22}=-1\Rightarrow f^1={\alpha }_3,f^2=-{\alpha }_2,f^3={\alpha }_1.$$

Maxwell Equations (22) and (23) take the form:

$$\dot{{\alpha }_a}={\beta }^b{n}_{ab}.$$

(76)

$${\alpha }_1{n}_{33}-{\alpha }_2{n}_{23}+{\alpha }_3{n}_{13}=-\dot{{\beta }^1},$$

(77)

$${\alpha }_1{n}_{23}-{\alpha }_2{n}_{22}+{\alpha }_3{n}_{12}=\dot{{\beta }^2},$$

$${\alpha }_1{n}_{13}-{\alpha }_2{n}_{12}+{\alpha }_3{n}_{11}=-\dot{{\beta }^3}.$$

These equations are symmetric with respect to the substitution ${\xi }_1^{\alpha }\iff {\xi }_2^{\alpha }$, Therefore, given this symmetry, for a complete classification it is sufficient to consider options:

$\mathbf{B}{\beta }^2\ne 0,\mathbf{C}{\beta }^2=0,{\beta }^1\ne 0.$ Option $\mathbf{D}({\beta }^a=0$) does not need to be considered, since for non-solvable groups $G_3\left(VIII\right),G_3\left(IX\right)$ it leads to a zero electromagnetic field.

Option B. ${\beta }^2\ne 0.$ The Equation (77) can be reduced to the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}(\dot{{\alpha }_1}-{\beta }^1{n}_{11}-{\beta }^3{n}_{13}),$$

(78)

$$n_{23}={\displaystyle \frac{1}{{\beta }^2}}(\dot{{\alpha }_3}-{\beta }^1{n}_{13}-{\beta }^3{n}_{33}).$$

(79)

$$n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}(\dot{{\alpha }_2}{\beta }^2-\dot{{\alpha }_3}{\beta }^3-\dot{{\alpha }_1}{\beta }^1+{{\beta }^1}^2{n}_{11}+2{\beta }^1{{\beta }^3}^2{n}_{12}+{{\beta }^3}^2{n}_{33}).$$

(80)

Using relations (78)–(80), the remaining Maxwell equations can be represented as

$$n_{33}({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)+n_{13}({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)={\alpha }_2\dot{{\alpha }_3}-{\beta }^2\dot{{\beta }^1},$$

(81)

$$n_{13}({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)+n_{11}({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)={\alpha }_2\dot{{\alpha }_1}-{\beta }^2\dot{{\beta }^3},$$

(82)

$${{\beta }^2}^2+{{\alpha }_2}^2-2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)=c=const.$$

(83)

Variant ${\mathbf{B}}_{\mathbf{1}}.$$({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)\ne 0.$ From Equations (81) and (82) it follows:

$$n_{13}={\displaystyle \frac{1}{{\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3}}({\beta }^2\dot{{\beta }^3}+{\alpha }_2\dot{{\alpha }_1}-n_{11}({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)),$$

(84)

$$n_{33}={\displaystyle \frac{1}{{({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)}^2}}(({\beta }^2\dot{{\beta }^1}+{\alpha }_2\dot{{\alpha }_3})({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)-({\beta }^2\dot{{\beta }^3}+{\alpha }_2\dot{{\alpha }_1})({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)+n_{11}{({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)}^2).$$

The solution to Equation (83) can be written as

$${\beta }^2=\varsigma \sqrt{2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)-{{\alpha }_2}^2+c}.$$

Variant ${\mathbf{B}}_{\mathbf{2}}.n_{33}$ is an arbitrary function ⇒

$${\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3=0.$$

(85)

Let us designate:

$${\alpha }_1=\gamma {\alpha }_2,{\beta }^3=-\gamma {\beta }^2.$$

Then solutions of Equations (80)–(82) have the form:

$$n_{11}={\displaystyle \frac{\gamma }{{\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1}}({\alpha }_1\dot{{\alpha }_1}+{\beta }^3\dot{{\beta }^3}),n_{13}={\displaystyle \frac{\gamma }{{\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1}}({\alpha }_1\dot{{\alpha }_3}+{\beta }^3\dot{{\beta }^1}).$$

The function $\gamma$ can been found from Equations (82) and (85) and has the form:

$$\gamma =\varsigma \sqrt{{\displaystyle \frac{c+2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)}{{{\alpha }_1}^2+{{\beta }^3}^2}}}$$

Variant ${\mathbf{B}}_{\mathbf{3}}.n_{11},n_{13}$ are arbitrary functions ⇒ the functions ${\alpha }_a,{\beta }^a$ are determined from the system of equations:

$$({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)={\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1=0.$$

(86)

$${\beta }^2\dot{{\beta }^1}+{\alpha }_2\dot{{\alpha }_3}=0,{\beta }^2\dot{{\beta }^3}+{\alpha }_2\dot{{\alpha }_1}=0,$$

(87)

$${{\beta }^2}^2-{{\alpha }_2}^2+2{\alpha }_1{\alpha }_3-2{\beta }^1{\beta }^3=c=const.$$

(88)

Let us designate:

$${\alpha }_1={\gamma }_3{\alpha }_2,{\alpha }_3={\gamma }_1{\alpha }_2,{\beta }^1=-{\gamma }_1{\beta }_2,{\beta }^3=-{\gamma }_3{\beta }_2.$$

Then the Equation (87) takes the form:

$$\dot{{\gamma }_p}({{\alpha }_2}^2+{{\beta }^2}^2)+{\gamma }_p({\alpha }_2\dot{{\alpha }_2}+{\beta }^2\dot{{\beta }^2})=0,p=1,3.$$

(89)

There are two nonequivalent solutions of the system of Equation (89):

• ${\gamma }_q=0\Rightarrow {\alpha }_q={\beta }^q=0,{\alpha }_2=pcos\omega ,{\beta }^2=psin\omega$
• ${\gamma }_p=c_p\gamma ={\displaystyle \frac{c_p}{\sqrt{{{\alpha }_2}^2+{{\beta }^2}^2}}},{\alpha }_1=c_3\gamma {\alpha }_2,{\alpha }_3=c_1\gamma {\alpha }_2,{\beta }^1=-c_1\gamma {\beta }_2,{\beta }^3=-c_3\gamma {\beta }_2.$

Option C. ${\beta }_2=0,{\beta }_3\ne 0.$

The Equation (77) can be reduced to the form:

$$n_{13}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_1}-{\beta }^1{n}_{11}),$$

(90)

$$n_{23}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_2}-{\beta }^1{n}_{12}).$$

(91)

$$n_{33}={\displaystyle \frac{1}{{{\beta }^3}^2}}(\dot{{\alpha }_3}{\beta }^3-\dot{{\alpha }_1}{\beta }^1+n_{11}{{\beta }^1}^2).$$

(92)

Using relations (90) and (91), the remaining equations from the Maxwell system of equations can be represented as

$$n_{11}({\alpha }_1{\beta }^1-{\alpha }_3{\beta }^3)+n_{12}{\alpha }_2{\beta }^3={\beta }^3\dot{{\beta }^3}+{\alpha }_1\dot{{\alpha }_1},$$

(93)

$$-n_{12}({\alpha }_3{\beta }^3-{\alpha }_1{\beta }^1)+n_{22}{\alpha }_2{\beta }^3={\alpha }_1\dot{{\alpha }_2},$$

(94)

$$2({\beta }^1{\beta }^3+{\alpha }_1{\alpha }_3)-{{\alpha }_2}^2=c=const.$$

(95)

Let us present all nonequivalent solutions.

Variant $C_1{\alpha }_2\ne 0.$ From Equations (93), (94) it follows:

$$n_{22}={\displaystyle \frac{1}{{\left({\alpha }_2{\beta }^3\right)}^2}}({\alpha }_2{\beta }^3({\alpha }_1\dot{{\alpha }_2}-{\beta }^3\dot{{\beta }^2})+({\alpha }_3{\beta }^3-{\alpha }_1{\beta }^1)({\alpha }_1\dot{{\alpha }_1}+{\beta }^3\dot{{\beta }^3})+n_{11}{({\alpha }_3{\beta }^3-{\alpha }_1{\beta }^1)}^2),$$

(96)

$$n_{12}={\displaystyle \frac{1}{{\alpha }_2{\beta }^3}}({\alpha }_1\dot{{\alpha }_1}+{\beta }^3\dot{{\beta }^3}+n_{11}({\alpha }_3{\beta }^3+{\alpha }_1{\beta }^1)),$$

(97)

$n_{11}$ is an arbitrary function. Solution of Equation (95) can be present in the form:

$${{\alpha }_2}^2=\varsigma \sqrt{2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)+c}.$$

Variant ${\mathbf{C}}_{\mathbf{2}}$. ${\alpha }_2=0,{\alpha }_3{\beta }^3-{\alpha }_1{\beta }^1\ne 0.$ Solution has the form:

$$n_{11}={\displaystyle \frac{{\beta }^3\dot{{\beta }^3}+{\alpha }_1\dot{{\alpha }_1}}{{\alpha }_1{\beta }^1-{\alpha }_3{\beta }^3}},n_{12}=0,{\beta }^1={\displaystyle \frac{c-{\alpha }_1{\alpha }_3}{{\beta }^3}},$$

Variant ${\mathbf{C}}_{\mathbf{3}}$.${\alpha }_1=pcos\omega ,{\alpha }_2=0,{\alpha }_3=ccos\omega ,{\beta }^1=csin\omega {\beta }^2=0,{\beta }^3=psin\omega$, $n_{11},n_{12}$ are arbitrary functions.

6.2. Group $G_3$(IX)

The Killing vector fields ${\xi }_a^{\alpha }$ and the canonical frame vector fields ${\zeta }_a^{\alpha }$:

$${\xi }_a^{\alpha }={\delta }_1^{\alpha }{\delta }_a^1+{\delta }_a^2({\delta }_3^{\alpha }cos{u}_2+{\displaystyle \frac{sin{u}^2}{cos{u}^3}}({\delta }_1^{\alpha }+{\delta }_2^{\alpha }sin{u}^3))+{\delta }_a^3{\xi }_{2,2}^{\alpha };$$

(98)

$${\zeta }_{\left(a\right)}^{\alpha }={\delta }_1^{\alpha }{\delta }_a^1+{\delta }_a^2(({\delta }_1^{\alpha }{\displaystyle \frac{sin{u}^3}{cos{u}^3}}+{\delta }_2^{\alpha }{\displaystyle \frac{1}{cos{u}^3}})sin{u}^1+{\delta }_3^{\alpha }cos{u}^1)+{\delta }_a^3{\zeta }_{2,2}.$$

(99)

Hence:

$$f^a={\alpha }_a,C^{11}=C^{22}=C^{33}=1,$$

and Maxwell equations have the form:

$${\alpha }_1{n}_{a1}+{\alpha }_2{n}_{a2}+{\alpha }_3{n}_{a3}=-\dot{{\beta }^a},$$

(100)

$$\dot{{\alpha }_a}={\beta }^b{n}_{ab}.$$

(101)

Equations (100) and (101) are symmetric with respect to permutations of the vectors of the canonical frame. Therefore, for a complete classification, it is sufficient to consider the variant ${\beta }^3\ne 0.$ In this case, the Equation (22) can be reduced to the form:

$$n_{13}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_1}-{\beta }^1{n}_{11}-{\beta }^2{n}_{12}),$$

(102)

$$n_{23}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_2}-{\beta }^1{n}_{12}-{\beta }^2{n}_{22}).$$

(103)

$$n_{33}={\displaystyle \frac{1}{{{\beta }^3}^2}}(\dot{{\alpha }_3}{\beta }^3-\dot{{\alpha }_1}{\beta }^1-\dot{{\alpha }_2}{\beta }^2+{{\beta }^1}^2{n}_{11}+2{\beta }^1{\beta }^2{n}_{12}+{{\beta }^2}^2{n}_{33}).$$

(104)

Using relations (102) and (103), the remaining Maxwell equations can be represented as

$$n_{11}({\alpha }_3{\beta }^1-{\alpha }_1{\beta }^3)+n_{12}({\alpha }_3{\beta }^2-{\alpha }_2{\beta }^3)={\beta }^3\dot{{\beta }^1}+{\alpha }_3\dot{{\alpha }_1},$$

(105)

$$n_{12}({\alpha }_3{\beta }^1-{\alpha }_1{\beta }^3)+n_{22}({\alpha }_3{\beta }^2-{\alpha }_2{\beta }^3)={\beta }^3\dot{{\beta }^2}+{\alpha }_3\dot{{\alpha }_2},$$

(106)

$${{\alpha }_1}^2+{{\alpha }_2}^2+{{\alpha }_3}^2+{{\beta }_1}^2+{{\beta }_2}^2+{{\beta }_3}^2=c^2=const.$$

(107)

In the case when $({\alpha }_1{\beta }^3-{\alpha }_3{\beta }^1)\ne 0,$ Equations (104) and (105) allow one to represent the solution in the form:

Variant 1.

$$n_{12}={\displaystyle \frac{1}{{\alpha }_3{\beta }^1-{\alpha }_1{\beta }^3}}({\beta }^2\dot{{\beta }^3}+{\alpha }_3\dot{{\alpha }_2}+n_{22}({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)),$$

(108)

$$n_{11}={\displaystyle \frac{{\beta }^3\dot{{\beta }^1}+{\alpha }_3\dot{{\alpha }_1}}{{\alpha }_3{\beta }^1-{\alpha }_1{\beta }^3}}+{\displaystyle \frac{({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)}{{({\alpha }_1{\beta }^3-{\alpha }_3{\beta }^1)}^2}}({\beta }^2\dot{{\beta }^3}+{\alpha }_3\dot{{\alpha }_2}+n_{22}({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)).$$

(109)

Here, $n_{22}$ is an arbitrary function, ${\alpha }_a,{\beta }^1,{\beta }^3$ obey the Equation (106), the solution of which can be represented as

$${\beta }^2=\varsigma \sqrt{{{\alpha }_1}^2+{{\alpha }_2}^2+{{\alpha }_3}^2+{{\beta }^1}^2+{{\beta }^3}^2+c^2}.$$

Variant 2. $({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)\ne 0$,

$$({\alpha }_1{\beta }^3-{\alpha }_3{\beta }^1)=0.$$

(110)

In this case, from Equations (106), (106) it follows that the functions $n_{12},n_{22}$ have the form:

$$n_{12}={\displaystyle \frac{{\beta }^3\dot{{\beta }^1}+{\alpha }_2\dot{{\alpha }_1}}{{\alpha }_3{\beta }^2-{\alpha }_2{\beta }^3}},n_{22}={\displaystyle \frac{{\beta }^3\dot{{\beta }^2}+{\alpha }_3\dot{{\alpha }_2}}{{\alpha }_3{\beta }^2-{\alpha }_2{\beta }^3}},$$

$n_{11}$ is an arbitrary function. Let us denote ${\beta }^1=\gamma {\beta }^3,{\alpha }_1=\gamma {\alpha }_3$, where $\gamma$ is an arbitrary function. The solution of the Equation (106), (110) can be represented as

$${\beta }^2=\varsigma \sqrt{{c^2-{\alpha }_2}^2-({{\alpha }_3}^2+{{\beta }^3}^2)(1+{\gamma }^2)}.$$

Variant 3. $n_{11},n_{22}$ are arbitrary functions,

$${\alpha }_1{\beta }^3-{\alpha }_3{\beta }^1)=({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)=0.$$

(111)

Using the same notations as in the second variant, one obtains:

$${\alpha }_1=\gamma {\beta }^1{\alpha }_2=\gamma {\beta }^2,{\alpha }_3=\gamma {\beta }^3.$$

The solution of the Equations (106), (111) can be represented as

$$\gamma =\varsigma \sqrt{{\displaystyle \frac{c^2}{{{\beta }_1}^2+{{\beta }_2}^2+{{\beta }_3}^2}}-1}$$

7. List of Results

This section presents all the solutions obtained. The results are presented in the next manner. A separate subsection is devoted to each group. All non-holonomic components of the metric tensor ($n_{ab}$) and electromagnetic potential (${\alpha }_a$) are listed. The sets of Killing vectors ${\xi }_a^{\alpha }$ and canonical frame vectors ${\zeta }_a^{\alpha }$ are presented in the Section 4, Section 5 and Section 6.

7.1. Solutions for the Group $G_3$(I)

1. 

$n_{a3}=c\dot{{\alpha }_a},$${\alpha }_a,n_{pq}$ are arbitrary functions, p, q = 1, 2.

7.2. Solutions for the Group $G_3$(II)

2. 

$n_{12},n_{22}$ are arbitrary functions,

$$n_{11}={\displaystyle \frac{1}{{\alpha }_1}}{\dot{\beta }}^1,n_{13}={\displaystyle \frac{{\dot{\alpha }}_1{\alpha }_1+{\beta }^1{\dot{\beta }}_1}{{\alpha }_1}},n_{23}={\dot{\alpha }}_2-n_{12}{\beta }^1,n_{33}={\dot{\alpha }}_3-{\beta }^1{\displaystyle \frac{{\dot{\alpha }}_1{\alpha }_1+{\beta }^1{\dot{\beta }}_1}{{\alpha }_1}}.$$

3. 

$n_{11},n_{11},n_{22}$ are arbitrary functions,

$${\alpha }^1=0,{\beta }^1=c,n_{13}=-c{n}_{11},n_{23}={\dot{\alpha }}_2-c{n}_{12},n_{33}={\dot{\alpha }}_3+c^2$$

4. 

$n_{22},n_{23},n_{23}$ are arbitrary functions,

$${\alpha }_1=asin\omega ,{\beta }^1=acos\omega n_{11}=\dot{\omega },n_{12}={\displaystyle \frac{\dot{{\alpha }_2}}{acos\omega }}{n}_{13}={\displaystyle \frac{\dot{{\alpha }_3}}{acos\omega }}.$$

7.3. Solutions for the Group $G_3$(III)

Option $\mathbf{B}$. Functions $n_{2a}$ have the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-c{n}_{11}),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_{1}c+c^2{n}_{11}),n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{13}c).$$

The rest functions are listed below ($\dot{a}=\dot{a_a}=0$).

5. 

$n_{11},n_{13},n_{33}$ are arbitrary functions $c=0,{\alpha }_1=asin\omega ,{\beta }^2=acos\omega ,n_{12}=\dot{\omega }$.

6. 

$n_{13},n_{33}$ are arbitrary functions.

$$n_{11}={\displaystyle \frac{\dot{{\alpha }_1}{\alpha }_1+{\beta }^2\dot{{\beta }^2}}{c{\alpha }_1}},n_{12}=-{\displaystyle \frac{\dot{{\beta }^2}}{{\alpha }_1}}.$$

7. 

$n_{11},n_{13},n_{33}$ are arbitrary functions, ${\alpha }_1=0,{\beta }^2=1.$

Option $\mathbf{C}{n}_{22},n_{23},n_{33}$ are arbitrary functions, $n_{1a}=\dot{{\alpha }_a}.$

8. 

${\alpha }_2=const,$

9. 

${\alpha }_1=0$.

Option $\mathbf{D}$ ${\alpha }_a=const$.

10. 

${\alpha }_1=0.$

11. 

$n_{12}=0.$

7.4. Solutionsfor the Group $G_3$(IV)

Option $\mathbf{B}$ The components $n_{2q}$ have the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-n_{11}{\beta }^1),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+n_{11}{{\beta }^1}^2).n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{13}{\beta }^1),$$

$n_{13},n_{33}$ are arbitrary functions. The rest functions are listed below.

12. 

$n_{11}={\displaystyle \frac{{\dot{\beta }}^2}{{\alpha }_2}},{\beta }_1={\beta }_2(c-ln{\beta }_2).$

13. 

$n_{11}=\dot{\sigma }{\alpha }_1=exp\sigma cos\omega ,{\beta }^2=exp\sigma sin\omega ,{\alpha }_2=\sigma exp\sigma cos\omega ,{\beta }^1=-\sigma exp\sigma sin\omega .$

14. 

$n_{11}={\displaystyle \frac{\dot{\sigma }}{\dot{{\gamma }_1}+\dot{{\gamma }_2}}}\sqrt{(\dot{\sigma }+\dot{{\gamma }_1})(\dot{{\gamma }_2}-\dot{\sigma })},{\alpha }_2={\gamma }_2{\alpha }_1,{\alpha }_1=exp\sigma \sqrt{{\displaystyle \frac{\dot{\sigma }+\dot{{\gamma }_1}}{\dot{{\gamma }_1}+\dot{{\gamma }_2}}}},$

$${\beta }^2=exp\sigma \sqrt{{\displaystyle \frac{\dot{{\gamma }_2}-\dot{\sigma }}{\dot{{\gamma }_2}+\dot{{\gamma }_1}}}},{\beta }^1={\gamma }_1{\beta }^2.$$

15. 

$n_{11}$ is an arbitrary function,

$${\alpha }_1=(a_1+a_2)cos\omega ,{\beta }^2=(a_1+a_2)sin\omega ,{\alpha }_2=-a_{2}cos\omega ,{\beta }^1=-a_{1}sin\omega .$$

Option C $n_{1a}={\displaystyle \frac{{\alpha }_a}{{\beta }^1}}.$

16. 

$n_{22}$ is an arbitrary function, $n_{11}=0,n_{12}=\dot{\omega },{\alpha }_1=0,{\alpha }_2=acos\omega ,{\beta }^1=asin\omega .$

17. 

$n_{11}$ is an arbitrary function,

$$n_{22}={\displaystyle \frac{1}{{\beta }^1{\alpha }_1}}({\dot{\alpha }}_2{\alpha }_2+{\dot{\beta }}^1{\beta }^1+\dot{{\alpha }_2}{\alpha }_1).{\alpha }_2={\alpha }_1(c+ln{\alpha }_1).$$

Option D ${\beta }^a=0,{\alpha }_a=a_a=const.$

18. 

$n_{11}=n_{12}=0,a_1=0.$

19. 

$n_{11}$ is an arbitrary function, $n_{12}=n_{11}(1+c),n_{22}=n_{11}{(1+c)}^2,a_2=c{a}_1.$

7.5. Solutions for the Groups $G_3$ (V) and $G_3$ (VI)

Option $\mathbf{B}$. Functions $n_{2a}$ have the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-{\beta }^1{n}_{11}),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+{{\beta }^1}^2{n}_{11}),n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{13}{\beta }^1).$$

The rest functions are listed below.

Variant ${\mathbf{B}}_{\mathbf{1}}.n_{11}={\displaystyle \frac{1}{{\beta }^1{\alpha }_1+q{\beta }^2{\alpha }_2}}({\alpha }_1{\dot{\alpha }}_1+{\beta }^2{\dot{\beta }}^2)$

20. 

${\alpha }_1=0,{\beta }^1=c{\left({\beta }^2\right)}^q.$

21. 

${\alpha }_2=c_2{\left({\alpha }_1\right)}^q,{\beta }^1=c_1{\left({\beta }^2\right)}^q.$

22. 

${\alpha }_1={\beta }^2{\left({\displaystyle \frac{\dot{\gamma }}{\dot{\omega }}}\right)}^{{\displaystyle \frac{1}{q+1}}},{\alpha }_2=\omega {\left({\beta }^2\right)}^q\left({\displaystyle \frac{\dot{\gamma }}{\dot{\omega }}}\right),{\beta }_1=q\gamma {\left({\beta }^2\right)}^q.$

Variant ${\mathbf{B}}_{\mathbf{2}}.n_{11}$ is an arbitrary function.

23. 

${\beta }^1=-q{a}_{1}sin\omega ,{\alpha }_2=a_{1}cos\omega ,{\alpha }_1=acos\omega ,{\beta }^2=asin\omega .$

Option D. ${\beta }^a=0,{\alpha }_a=const$.

24. 

$n_{12}=n_{22}=a_2=0.$

7.6. Solutions for the Group $G_3\left(VII\right)$

Option $\mathbf{B}$. Functions $n_{2a}$ have the form:

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_1-{\beta }^1{n}_{11}),n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}({\dot{\alpha }}_2{\beta }^2-{\dot{\alpha }}_1{\beta }^1+{{\beta }^1}^2{n}_{11}),n_{23}={\displaystyle \frac{1}{{\beta }^2}}({\dot{\alpha }}_3-n_{13}{\beta }^1).$$

$n_{13},n_{33}$ are arbitrary functions. The rest functions are listed below.

Variant ${\mathbf{B}}_{\mathbf{1}}{n}_{11}={\displaystyle \frac{1}{({\beta }^1{\alpha }_2+{\beta }^2(q{\alpha }_2-{\alpha }_1))}}({\alpha }_2{\dot{\alpha }}_1+{\beta }^2({\dot{\beta }}^1+q{\dot{\beta }}^2)).$

25. 

${\alpha }_1=a_1,{\beta }^2=a_2,{\alpha }_2=acos\omega ,{\beta }^1=asin\omega .$

26. 

${\Phi }^2=({\left(\dot{\gamma }\dot{{\beta }^2}\right)}^2+({\dot{{\alpha }_1}}^2+{\dot{{\beta }^2}}^2)({\dot{{\alpha }_1}}^2(2q\gamma -{{\alpha }_1}^2-{{\beta }^2}^2)-{\dot{\gamma }}^2),$

$${\alpha }_2={\displaystyle \frac{\varsigma }{\dot{{\alpha }_1}}}\sqrt{\dot{\gamma }+{\beta }_1\dot{{\beta }^2}},{\beta }_1={\displaystyle \frac{1}{{\left(\dot{{\alpha }_1}\right)}^2+{\left(\dot{{\beta }^2}\right)}^2}}(\dot{\gamma }\dot{{\beta }^2}+\Phi ).$$

Variant ${\mathbf{B}}_{\mathbf{2}}$ $n_{11}$ is an arbitrary function.

27. 

${\alpha }_1=asin\omega ,{\alpha }_2=a_{1}sin\omega ,{\beta }_1=acos\omega (p-qb),{\beta }_2=a_{1}cos\omega .$

28. 

${\alpha }_1={\alpha }_2=0,{\beta }^1=const,{\beta }^2=const.$

Option $\mathbf{C}.{\beta }^2=0,n_{1a}={\displaystyle \frac{\dot{\alpha }}{{\beta }^1}}.$

Variant ${\mathbf{C}}_{\mathbf{1}}{\alpha }_2\ne 0,n_{22}=(q{\alpha }_2-{\alpha }_1){\displaystyle \frac{\dot{{\alpha }_2}}{{\beta }_1{\alpha }_2}},$

29. 

${\alpha }_2={\displaystyle \frac{\dot{\gamma }}{\dot{{\alpha }_1}}},{\beta }_1={\displaystyle \frac{\varsigma }{\dot{{\alpha }_1}}}\sqrt{{\dot{{\alpha }_1}}^2(2q\gamma -{{\alpha }_1}^2)-{\dot{\gamma }}^2}.$

30. 

${\alpha }_1=a_1,{\alpha }_2=acos\omega ,{\beta }^1=asin\omega .$

Variant ${\mathbf{C}}_{\mathbf{2}}{\alpha }_2=0,n_{22}$ is an arbitrary function,

31. 

${\alpha }_1=acos\omega ,{\beta }^1=asin\omega .$

Variant $\mathbf{D}{\beta }^a=0,{\alpha }_b=a_b=const$.

32. 

$n_{11}=n_{12}=0,a_2=0,$.

33. 

$a_1=a_2=0$.

34. 

$a_1=c{a}_2,a_2\ne 0,n_{12}=n_{11}(q-1),n_{22}=n_{11}{(q-1)}^2$.

7.7. Solutions for the Group $G_3\left(VIII\right)$

Option $\mathbf{B}$. Functions $n_{2a}$ have the form (34):

$$n_{12}={\displaystyle \frac{1}{{\beta }^2}}(\dot{{\alpha }_1}-{\beta }^1{n}_{11}-{\beta }^1{n}_{13}),n_{23}={\displaystyle \frac{1}{{\beta }^2}}(\dot{{\alpha }_3}-{\beta }^1{n}_{13}-{\beta }^3{n}_{33}).$$

$$n_{22}={\displaystyle \frac{1}{{{\beta }^2}^2}}(\dot{{\alpha }_2}{\beta }^2-\dot{{\alpha }_3}{\beta }^3-\dot{{\alpha }_1}{\beta }^1+{{\beta }^1}^2{n}_{11}+2{\beta }^1{{\beta }^3}^2{n}_{12}+{{\beta }^3}^2{n}_{33}).$$

35. 

$$n_{13}={\displaystyle \frac{1}{{\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3}}({\beta }^2\dot{{\beta }^3}+{\alpha }_2\dot{{\alpha }_1}-n_{11}({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)),$$

$$n_{33}={\displaystyle \frac{1}{{({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)}^2}}(({\beta }^2\dot{{\beta }^1}+{\alpha }_2\dot{{\alpha }_3})({\alpha }_1{\beta }^2+{\alpha }_2{\beta }^3)-({\beta }^2\dot{{\beta }^3}+{\alpha }_2\dot{{\alpha }_1})({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)+$$

$$n_{11}{({\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1)}^2),{\beta }^2=\varsigma \sqrt{2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)-{{\alpha }_2}^2+c}.$$

36. 

$n_{33}$ is an arbitrary function.

$${\alpha }_1=\gamma {\alpha }_2,{\beta }^3=-\gamma {\beta }^2.n_{11}={\displaystyle \frac{\gamma }{{\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1}}({\alpha }_1\dot{{\alpha }_1}+{\beta }^3\dot{{\beta }^3}),$$

$$n_{13}={\displaystyle \frac{\gamma }{{\alpha }_3{\beta }^2+{\alpha }_2{\beta }^1}}({\alpha }_1\dot{{\alpha }_3}+{\beta }^3\dot{{\beta }^1}),\gamma =\varsigma \sqrt{{\displaystyle \frac{c+2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)}{{{\alpha }_1}^2+{{\beta }^3}^2}}}$$

37. 

$n_{11},n_{13}$ are arbitrary functions, ${\alpha }_1={\alpha }_3={\beta }^1={\beta }^3=0,{\alpha }_2=acos\omega ,{\beta }^2=asin\omega .$

38. 

$n_{11},n_{13}$ are arbitrary functions,

$${\gamma }_p=c_p\gamma ={\displaystyle \frac{c_p}{\sqrt{{{\alpha }_2}^2+{{\beta }^2}^2}}},{\alpha }_1=c_3\gamma {\alpha }_2,{\alpha }_3=c_1\gamma {\alpha }_2,{\beta }^1=-c_1\gamma {\beta }_2,{\beta }^3=-c_3\gamma {\beta }_2.$$

Option C. ${\beta }_2=0$,

$$n_{13}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_1}-{\beta }^1{n}_{11}),n_{23}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_2}-{\beta }^1{n}_{12}).n_{33}={\displaystyle \frac{1}{{{\beta }^3}^2}}(\dot{{\alpha }_3}{\beta }^3-\dot{{\alpha }_1}{\beta }^1+n_{11}{{\beta }^1}^2).$$

39. 

$n_{11}$ is an arbitrary function,

$$n_{22}={\displaystyle \frac{1}{{\left({\alpha }_2{\beta }^3\right)}^2}}({\alpha }_2{\beta }^3({\alpha }_1\dot{{\alpha }_2}-{\beta }^3\dot{{\beta }^2})+({\alpha }_3{\beta }^3-{\alpha }_1{\beta }^1)({\alpha }_1\dot{{\alpha }_1}+{\beta }^3\dot{{\beta }^3})+n_{11}{({\alpha }_3{\beta }^3-{\alpha }_1{\beta }^1)}^2),$$

$$n_{12}={\displaystyle \frac{1}{{\alpha }_2{\beta }^3}}({\alpha }_1\dot{{\alpha }_1}+{\beta }^3\dot{{\beta }^3}+n_{11}({\alpha }_3{\beta }^3+{\alpha }_1{\beta }^1)),{{\alpha }_2}^2=\varsigma \sqrt{2({\alpha }_1{\alpha }_3+{\beta }^1{\beta }^3)+c}.$$

40. 

$n_{22}$ is an arbitrary function, $n_{11}={\displaystyle \frac{{\beta }^3\dot{{\beta }^3}+{\alpha }_1\dot{{\alpha }_1}}{{\alpha }_1{\beta }^1-{\alpha }_3{\beta }^3}},n_{12}=0,{\alpha }_2=0,{\beta }^1={\displaystyle \frac{c-{\alpha }_1{\alpha }_3}{{\beta }^3}}.$

41. 

$n_{11},n_{12},n_{22}$ are arbitrary functions,

$${\alpha }_1=acos\omega ,{\alpha }_2=0,{\alpha }_3=a_{1}cos\omega ,{\beta }^1=a_{1}sin\omega {\beta }^2=0,{\beta }^3=asin\omega .$$

7.8. Solutions for the Group $G_3\left(IX\right)$

Functions $n_{3a}$ have the form:

$$n_{13}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_1}-{\beta }^1{n}_{11}-{\beta }^2{n}_{12}),n_{23}={\displaystyle \frac{1}{{\beta }^3}}(\dot{{\alpha }_2}-{\beta }^1{n}_{12}-{\beta }^2{n}_{22}).$$

$$n_{33}={\displaystyle \frac{1}{{{\beta }^3}^2}}(\dot{{\alpha }_3}{\beta }^3-\dot{{\alpha }_1}{\beta }^1-\dot{{\alpha }_2}{\beta }^2+{{\beta }^1}^2{n}_{11}+2{\beta }^1{\beta }^2{n}_{12}+{{\beta }^2}^2{n}_{33}).$$

The rest functions are listed below.

42. 

$n_{22}$ is an arbitrary function.

$$n_{12}={\displaystyle \frac{1}{{\alpha }_3{\beta }^1-{\alpha }_1{\beta }^3}}({\beta }^2\dot{{\beta }^3}+{\alpha }_3\dot{{\alpha }_2}+n_{22}({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)),$$

$$n_{11}={\displaystyle \frac{{\beta }^3\dot{{\beta }^1}+{\alpha }_3\dot{{\alpha }_1}}{{\alpha }_3{\beta }^1-{\alpha }_1{\beta }^3}}+{\displaystyle \frac{({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)}{{({\alpha }_1{\beta }^3-{\alpha }_3{\beta }^1)}^2}}({\beta }^2\dot{{\beta }^3}+{\alpha }_3\dot{{\alpha }_2}+n_{22}({\alpha }_2{\beta }^3-{\alpha }_3{\beta }^2)),$$

$${\beta }^2=\varsigma \sqrt{c^2-{{\alpha }_1}^2-{{\alpha }_2}^2-{{\alpha }_3}^2-{{\beta }^1}^2-{{\beta }^3}^2}.$$

43. 

$n_{11}$ is an arbitrary function,

$$n_{12}={\displaystyle \frac{{\beta }^3\dot{{\beta }^1}+{\alpha }_2\dot{{\alpha }_1}}{{\alpha }_3{\beta }^2-{\alpha }_2{\beta }^3}},n_{22}={\displaystyle \frac{{\beta }^3\dot{{\beta }^2}+{\alpha }_3\dot{{\alpha }_2}}{{\alpha }_3{\beta }^2-{\alpha }_2{\beta }^3}},$$

$${\beta }^1=\gamma {\beta }^3,{\alpha }_1=\gamma {\alpha }_3,{\beta }^2=\varsigma \sqrt{{c^2-{\alpha }_2}^2-({{\alpha }_3}^2+{{\beta }^3}^2)(1+{\gamma }^2)}.$$

44. 

$n_{11},n_{22}$ are arbitrary functions,

$${\alpha }_1=\gamma {\beta }^1{\alpha }_2=\gamma {\beta }^2,{\alpha }_3=\gamma {\beta }^3.\gamma =\varsigma \sqrt{{\displaystyle \frac{c^2}{{{\beta }_1}^2+{{\beta }_2}^2+{{\beta }_3}^2}}-1}.$$

8. Conclusions

Results obtained in this paper can be used to solve the problem of classifying exact solutions of the vacuum Einstein-Maxwell equations for homogeneous Petrov non-null spaces. For Stackel spaces this classification problem has been solved to a large extent (including in our papers; see, for example, [20] and the literature cited there). However, for Petrov spaces this problem is at an early stage of solution. A complete classification has been carried out only for spaces with Abelian group of motions $G_3\left(I\right)$ (see [42,43]. Let us note that non-null spaces with a three-parameter Abelian group of motions belong to both Petrov spaces and Stäckel spaces of type (3.0)). The classification of admissible electromagnetic fields and exact solutions of Maxwell vacuum equations and the presentation of the results in a form convenient for use are the initial stage of the classification of electrovacuum spaces associated with homogeneous Petrov spaces.