On Rational Solutions of Dressing Chains of Even Periodicity

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction identifies rational solutions with points on orbits of fundamental shift operators acting on first-order polynomial solutions derived for dressing chains of even periodicity. We also obtain conditions for the existence of special function solutions that occur for a special class of first-order polynomial solutions. For the special case of the N=4 dressing chain equations the method yields all the known rational solutions of Painlev\'e V equation. They are obtained through action of shift operators on the two independent first-order polynomial solutions. The formalism naturally extends to N=6 and beyond as shown in the paper.


Introduction and background information
Painlevé equations form a class of second order nonlinear differential equations with solutions that have no movable critical singularities in the complex plane, see e.g. [1]. Although this mathematical property motivated the discovery of Painlevé equations, in relatively short time, these equations made astonishing impact on several fields inside and outside mathematics. A long and incomplete list of affected topics and models includes correlation functions of the Ising model, random matrix theory, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear and fiber optics, and Bose-Einstein condensation. Special solutions, such as rational solutions, turned out to be important in these applications and various methods were applied in their study. To provide a systematic approach to the study of rational solutions we here utilize the dressing chain and its connection to Painlevé equations. The dressing chain was derived by applying Darboux transformations to the spectral problem of second order differential equation [2]. Specifically, let us consider a sequence of second order differential operators L n connected via first order Darboux transformations : (∂ z −j n )L n = (L n−1 +α n )(∂ z −j n ), where α n is a constant. Such symmetry is realized for L n = (∂ z + j n )(∂ z − j n ) + α n = (∂ z − j n+1 )(∂ z + j n+1 ) , (1.1) with L n defined by products of two first order differential operators with their orders being interchanged when going from n to n + 1. Comparing the two alternative expressions for L n in equation (1.1) we obtain the nonlinear lattice equations [2] : (j n + j n+1 ) z = −j 2 n + j 2 n+1 + α n , n = 1, . . ., N, j N +i = j i , (1.2) made finite by imposing the periodic boundary condition j N +i = j i . We refer to system (1.2) as a system of dressing chain equations of N -periodicity. Such system possesses many important properties. For N = 3, it has been shown [2] that it passes the Kovalevskaya-Painlevé test and its equivalence to Painlevé IV equation has also been established [3,2]. For higher N the system is equivalent to A N −1 invariant Painlevé equations [3,4] and this equivalence will be utilized in this paper to construct and study rational solutions of Painlevé equations in the context of underlying periodic dressing chains. Quite recently the N cyclic dressing chain was also obtained in the self-similarity limit of the second flow of sl(N ) mKdV hierarchy [5].
As we will now show the system (1.2) requires different treatments depending on whether N is odd or even. This becomes evident when we consider a regular sum N n=1 (j n +j n+1 ) z and an alternating sum N n=1 (−1) n (j n +j n+1 ) z of derivatives of j n +j n+1 . Calculating a regular sum using the dressing equations (1.2) we obtain for both even and odd N the same expression for the integration constant on the right hand side. As long as N is odd calculating an alternating sum N n=1 (−1) n (j n + j n+1 ) z using the dressing equations (1.2) will reproduce the same condition as in (1.3). For even N the alternating sum N n=1 (−1) n (j n + j n+1 ) z is identically zero (positive and negative terms simply cancel). However the same expression calculated by plugging the right hand side of dressing equations (1.2) yields for e.g. N = 4 the expression j 2 1 + j 2 3 − j 2 2 − j 2 4 + 1 2 (−α 1 + α 2 − α 3 + α 4 ) . Thus the dressing chains of even periodicity require imposition of a new quadratic constraint or modification of the dressing chain formulation. Such modification was proposed in [6], where the authors put forward a system of dressing chain equations of even N = 4, 6, 8, . . . periodicity defined as : This structure is such that both regular and alternating sums of derivatives of j i + j i+1 give consistent answers when applied on the system (1.4): As shown in [6] such system can be obtained by Dirac reduction from N + 1 dressing chain (1.2) of odd periodicity.
The above equations as well as quantities Ψ and Φ are invariant under A N −1 Bäcklund transformations s i , i = 1, . . ., N [3] : when transformations (1.6) are accompanied by transformations of coefficients There are also two automorphisms π, ρ : it holds that the corresponding sum f n = j n + j n+1 =j n +j n+1 is unchanged. Such redefinition leads to a formal absorption of Ψ terms so that they are no longer explicit in the dressing equations rewritten in terms ofj n that satisfy equations (1.2) [6]. However such process introduces potential extra divergencies into an associated Sturm-Liouville problem. Throughout this paper we will work with (1.4) with a constant non-zero Ψ so that the polynomial seed solutions we will construct below will be free of divergencies. We present construction of rational and special function solutions for dressing chains of even periodicity. In this work rational solutions are identified with points on the orbits of fundamental shift operators (sometimes also referred to in the literature as translations) of the extended affine Weyl group A (1) N −1 acting on the first-order polynomial seed solutions. In particular for the seed solutions with all the components being equal to each other the construction yields rational solutions being ratios of Umemura polynomials [7]. The reduction procedure that yields special function solutions is outlined and is shown to reproduce rational solutions for appropriate values of the parameters of the underlying Riccati equations.
The presentation is organized as follows. In section 2, we obtain the first-order polynomial solutions of the dressing chain equations (1.4) with parameters α i , i = 1, . . ., N depending on one arbitrary variable and with a constant non-zero Ψ that ensures that the solution is polynomial.
In section 3, we establish connection between the dressing chain equations (1.4) and Hamiltonian formalism for N = 4, 6 that can easily be generalized to arbitrary even N . Essential for establishing this connection is ability to cast the dressing chain equations (1.4) as symmetric A (1) N −1 -invariant Painlevé equations as those given in equations (3.1) and (A.1) for N = 4, 6, respectively. We should point out that translating the system of equations depending on j i into formalism that is expressed entirely in terms of f i = j i +j j+1 is possible for even N thanks due to the presence of Ψ terms on the right hand sides of equations (1.4). This is in contrast to odd N dressing chains where j i and f i are always fully interchangeable. For N = 4 the Hamiltonian formalism of section 3 gives rise to Painlevé V equation as briefly reviewed in subsection 3.2. The first-order polynomial solutions in the setting of Hamiltonian formalism become the algebraic solutions of [8].
We are able to present a power series expansions of Hamiltonian variables p and q in subsection 3.4. We show how potential divergencies of power series solutions (that can not be absorbed in Ψ) can be removed by appropriate Bäcklund transformations. After removing the eventual simple poles from rational solutions by acting with the Bäcklund transformations we obtain rational solutions that are expandable in a series of positive powers of z and can be reproduced by actions of the shift operators as shown in the next section.
In section 4, we derive rational solutions for N = 4 by acting with shift operators on the first-polynomial solutions (2.2) and (2.3) to obtain all known cases listed in reference [9] that presented necessary and sufficient conditions for rational solutions of Painlevé V equation. Reference [10] showed how to act with shift operators on solutions (2.2) (expressed by tau functions) to obtain some of the cases of [9] (items I + II on page 13). For the first-order polynomial seed solutions (2.2) (with all the components j i equal to z/N ) the action of shift operators yields rational solutions expressed by Umemura polynomials [7,11] and we use the shift operators to derive the recurrence relations that determine these polynomials. Extending structure of seed solutions to include solutions (2.3) (where j i + j i+1 = 0 for some i) requires exclusion of those shift operators that are ill-defined when acting on such solutions as discussed in subsection 4.5. Those of the shift operators that are well-defined generate the remaining rational solutions from solutions (2.3), see item III on page 13. This new approach leads to a systematic and unified way to derive all rational Painlevé V solutions. Based on results for N = 4 we conjecture for all even N that all rational solutions are obtainable through actions of shift operators on first-order polynomial solutions.
In section 5, we provide explicit construction of special function solutions and rational solutions for N = 6. The rational solutions are always identified with orbits of the fundamental shift operators. For the seed solution with all components being equal or only one of the components being negative we are able to express the corresponding rational solutions by Umemura type of polynomials. Existence of special function solutions is established for the remaining cases with a sufficient number of constraints imposed on α i parameters to insure reduction of Hamiltonian equations to one single Riccati equation. For N = 6 case this happens for three independent constraints. However we also encounter hybrid situations with one single Riccati equation and one coupled quadratic (in q i , p i ) equation for some cases with two constraints. In such cases there exists a special function solution for only one of the variables. Interestingly, when α i parameters are associated with orbits of the shift operators we obtain closed expressions in terms of Whittaker functions that describe rational solutions for all underlying variables of the reduced system.
2 Preliminaries. The seed solutions as the firstorder polynomial solutions of even chains For simplicity we first carry out the discussion for N = 4 before proceeding to the case of N = 6 and making general comments about higher N cases.
We are looking for the first-order polynomial solutions to equation (1.4) of the type that satisfy the Φ = z condition. With such ansatz the quantity Ψ defined in (1.5) can only contain terms with z 2 or a constant. The terms quadratic in z can be absorbed in j i via (1.9) transformation. Thus without losing any generality we can assume that where we used that 4 i=1 α i = 2. One can easily see that the condition for Ψ not to contain z 2 for the polynomial solutions of the first-order amounts to j 2 n+1 − j 2 n = 0 on the right hand side of the dressing equations. Thus the solution must be j i = zc(ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 ) with ǫ i = ±1 and c a non-zero constant. Since Φ = z = 0 we must also have ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 = 0. This argument eliminates the case of two epsilons being negative, ǫ i = −1, ǫ j = −1, i = j, as this would violate Φ = 0. Therefore the only two independent (up to π) polynomial solutions are : Both solutions depend on only one free parameter a. The remaining first order polynomial solutions can be obtained by acting with π, π 2 and π 3 on solution (2.3) (recall that π 4 = 1 for N = 4 cyclicity and so π 3 = π −1 ). Note that in case of solution (2.3) the action of automorphism π is such that it simply moves the −1 term in expression for j i and zeros in expression for α i to the right. It is important to point out that there could be other potential solutions of the first-order polynomial type like for example j i = (z/2)(1, 0, 1, 0). However such solutions would involve z 2 terms in Ψ and could be transformed by the transformation (1.9) involving the z 2 part of Ψ to the solution (2.3) or its π variants. One can easily extend this analysis to higher N with Ψ and Φ defined in the definition (1.5). For the N = 6 first-order polynomial solutions we take : and obtain five different first-order polynomial solutions: since all these configurations seems to be distinct and can not be connected by permutation generated by π or multiples of π's. All the above solutions depend on one arbitrary parameter a. Note that For arbitrary even N with Φ = z, Ψ = 1 − N/2 k=1 α 2k−1 and an arbitrary variable a there will always be a fully symmetric solution: which is a fixed point of π 2 automorphism. The remaining solutions will have one and up to N/2 − 1 negative components j i = − z N with varying distance between the negative components. For example for only one negative component in the last position we get with j k = 0, α k = 0 for k = N − 1, N , and so on for solutions with more negative components.
One needs to point out that the first-order solutions (2.4)-(2.8) appeared also as simple rational solutions expressed in terms of f i = j i + j i+1 that give rise to other rational solutions via Bäcklund transformations in the framework of A 5 Painlevé equations (equivalent to N = 6 dressing chain equations) in reference [12].
3 Hamiltonian formalism and polynomial solutions 3

.1 Hamilton equations and their algebraic solutions
For N = 4 we will show how the first-order polynomial solutions (2.2) and (2.3) are equivalent to all algebraic solutions found for Painlevé V equation in [8]. These solutions will then serve as seeds of all rational solutions [9] of Painlevé V equation via shift transformations.
The above system of equations can be cast into a Hamiltonian system with with Hamilton equations The Hamilton equations (3.3) reproduce the N = 4 system of equations (3.1) after substitution (q, p) → (f 1 , f 2 , f 3 , f 4 ) such that The Bäcklund transformations (1.6) and automorphisms (1.8) are given in the setting of Hamilton equations (3.3) by where α 4 is understood as 2 − α 1 − α 2 − α 3 in terms of α i , i = 1, 2, 3 appearing in the Hamiltonian formalism.
For N = 6 we define the Hamiltonian formalism in terms of quantities: which satisfy equations One of advantages of variables q i , p i , i = 1, 2 is that they make expressions for Bäcklund transformations (1.6) more transparent. The actions of Bäcklund transformations on these variables are given by where we only listed those transformations that are not identities and each s i is accompanied by transformation (1.7) of α i . The automorphism π acts in this setting as follows: The first-order polynomial solutions (2.4)-(2.8) are expressed in terms of variables defined in relation (3.10) as the following solutions to Hamilton equations (3.11) : 14) q 1 = p 1 = z, q 2 = 2z, p 2 = 0 α i = (a, 2 − a, a, 0, −a, 0) , (3.15) q 1 = q 2 = p 1 = z, p 2 = 0 α i = (2 − a, a, 0, 0, 0, 0) , (3.16) We notice that the solution (3.13) is a fixed point of π 2 automorphism as it is obvious comparing with its form in expression (2.4).

Connection of N = 4 formalism to Painlevé V equation
It is well-known that equations (3.1) or (3.3) lead to Painlevé V equation. We will here establish this relation explicitly in order to relate the parameters of both theories. We first define w = q/z. Taking a derivative of the top equation in (3.3) and eliminating p z and p we obtain the second order equation We need two additional steps to cast equation (3.18) into a standard form of Painlevé V equation.
First we perform a change of variables z → t where t = ǫz 2 /2 then followed by a transformation y = w/(w − 1).
In terms of y equation (3.18) takes a form of standard Painlevé V equation Forδ to take a conventional value of − 1 2 we need ǫ 2 = 1.

Riccati solutions of equations (3.1)
Let us reduce equations (3.1) by setting either α 2 = 0, f 2 = 0, f 4 = z or α 3 = 0, f 3 = 0, f 1 = z. Using that f 3 = z − f 1 in the first case and f 4 = z − f 2 in the second case we can rewrite the remaining equations for in which we recognize Riccati equations [13]. Without losing generality we will discuss the solution for the case of i = 1 with the principal solution given in terms of Whittaker functions as The above expression becomes a rational function for at least one of the two parameters α 1 , α 3 being equal to a negative even integer, and the other equal to an arbitrary integer but not equal to the opposite of that negative even integer (α 1 + α 3 = 0) : For the special case α 1 = 0 = α 3 it holds that F 1 = 0. With the above conditions being satisfied the rational solutions occur for Painlevé parameters: Let us recall that since α 2 = 0 then ǫγ = −α 4 /2 = −(2 − α 1 − α 3 )/2 . Thus if α 1 = −2n, n ∈ Z + then we can rewrite α 3 as α 3 = 2(1 + n + ǫγ). If α 3 = −2n, n ∈ Z + then α 1 = 2(1 + n + ǫγ). Riccati equation (3.22) takes a more familiar look when we rewrite it in terms of a variable x = −z 2 /2 : To linearize this equation we set F i = w i x /w i and for brevity introduce coefficients b i = (α i + α i+2 )/2 and a i = α i /2. In this way we obtain the second-order Kummer's equation: We look for solutions of Kummer's equation denoted as U (a, b, x) that are polynomials in x of a finite, let us say n, degree. This occurs for a = −n and for a − b = −n − 1 for n = 0, 1, 2, 3. . . and in the latter case it holds that [14] : where (a) r is a Porchhammer symbol. We will connect this polynomial with the case of α 3 = 0 and a = α 2 /2, b = (α 2 + α 4 )/2 for α i = (α 1 +2n, −2n, 0, 2−α 1 ), which we will revisit later in equation (4.50) in subsection 4.5, where it will be obtained by an action of T −n 2 shift operator on polynomial solutions (2.3). For such values of a and b we will need to calculate which is a polynomial of degree n according to equation (3.25).

Power series representation of p and q variables
For N = 4 we will show that q = j 1 + j 2 , p = j 2 + j 3 can be represented by power series in odd powers of z and the results are (up to an action with π automorphism and its powers) The second case can be transformed by s 1 Bäcklund transformation to the previous case. Consider power series expansion j i = k i z −m + . . . with the first term being lowest power in z . Comparing both sides of equations (1.4) we notice that the lowest terms on the left and the right sides will be of the order where we use the expansion of Ψ in (1.5) in powers of z : For the terms on both sides of (3.27) to match and cancel each other we need to take m = 1 and set all Ψ (k) = 0, k < 0. In such case only Ψ 0 contributes to the above equation. Without losing generality we therefore adopt the expansion For expansion in (3.28) it follows that Next we will effectively work with the dressing equations (1.2) without Ψ to see whether solutions for j i = a i /z + b i + c i z will be such that the divergent terms can be absorbed in Ψ of equation (1.4) via transformation (1.9): On the z −2 level of such dressing equations one finds the following expressions: for each i = 1, 2, 3, 4. There are two independent solutions of the above equations: that all satisfy i a i = 0. There are other similar solutions that one can obtain from (3.32) by acting with π, π 2 , π 3 transformations to obtain other solutions like e.g. a i = (a, a − 1, 1 − a, −a) and a i = (−1 + a, 1 − a, −a, a). It therefore suffices to use below the solution (3.32). The top equation (3.31) is such that a i + a i+1 = 0 for every i = 1, 2, 3, 4. Such divergence can be absorbed by the transformation (1.9) with Ψ = 2a. In addition the divergent terms will be absent from expressions for p and q. The other solution (3.32) is such that either a 1 +a 2 = 0 or a 2 +a 3 = 0 ensuring Ψ −2 = 0 according to relation (3.29). However the divergent terms are such that they can not be removed the transformation (1.9) and the divergent terms will be present in expressions for p. Let us illustrate this by applying the transformation (1.9) with Ψ = −2(1+ a). This results in a i = (1 + 2a, −(1 + 2a), 0, 0). As we will show below such divergent terms can be removed by a Bäcklund transformation. The calculations done for N = 4 and N = 6 suggest that this is a general feature for all N . Now for solution (3.31) we obtain that the condition (3.29) for Ψ (−1) = −2((a 1 + a 2 )(b 2 + b 3 ) + (a 2 + a 3 )(b 1 + b 2 )) = 0 is satisfied automatically and accordingly b i can be chosen arbitrarily. For (3.32) and the other configurations that can be obtained from (3.32) by π, we obtain conditions Consider now z −1 level of the equations (1.4) without Ψ. With such redefined system one obtains on the z −1 level 0 = a i+1 b i+1 − a i b i . For the solutions in (3.31) and (3.32) we find that we can write b i = b(1, −1, 1, −1) and we can set b = 0 without losing any generality as the terms can be added or removed by the transformation (1.9). Similar conclusion can be obtained for other coefficients of terms with z to the even power: z 2k . Such terms will not contribute to q = j 1 + j 2 , p = j 2 + j 3 and we don't need to consider them in what follows.
Consider now z 0 levels of the equations (1.2) : . We first plug values for a i from (3.31) into the above equation to obtain using that i c i = 1. For a i given in (3.32) we find and We will now apply our results to Here for brevity we introduced c 12 = c 1 + c 2 given in equation (3.35). Explicit calculation gives It follows that the singular term in p in (3.36) can be removed by s 1 transformation : q → q, p → p + α 1 /q with which shows that the transformed p given by p + α 1 /q will no longer contain a singular term. Its power expansion will start with the term proportional to z and will only contain odd powers of z.
The initial position of the pole can be obviously moved from p to q by the π automorphism. This will lead to s 1 being transformed by π to other s i , which will remove the divergent terms. With this understanding we continue to consider the above configuration without any loss of generality. One can therefore effectively only consider the case of a i = a(1, −1, 1, −1) from (3.31) with q = c 12 z + e 12 z 3 + . . ., p = c 23 z + e 23 z 3 + . . . , Amazingly the first terms of a general expression for q, p agree with a general formula that reproduces all the cases of (3.5)-(3.9) for the corresponding values of α i . Let us illustrate all this by the following example.
Applying equations (3.30) and (3.33) to N = 6 we find that the number of solutions increased from two to three (up to an action of π automorphism) and they are given by: Note that from equations (3.33) we find c 1 +c 2 = α 1 /(1+2a) and c 3 +c 4 = −α 3 /(1+2a) where a is given in relations (3.42) and (3.43), respectively.
In case of solution (3.42) the expansion of p 1 starts with a pole Consequently the action of s 1 on p 1 removes the pole similarly to what we have seen for N = 4 case in expression (3.37).
In case of solution (3.43) both expansions of p i , i = 1, 2 will start with divergent terms: Consequently the action of s 3 from equation (3.12) on p 1 and p 2 will remove these divergencies. For those solutions that are obtained from solutions (3.42) or (3.43) by acting with automorphism π or its powers the divergencies will be removed by appropriate Bäcklund transformations that are conjugations of s 1 , s 3 , e.g. πs 1 π −1 , πs 3 π −1 etc.

Construction of Rational Solutions
In this section, we will describe a method to derive all rational solutions that are obtainable from the first-order polynomial solutions of dressing equations (1.4) via combined actions of fundamental shift operators T i , i = 1, . . ., N from (4.8). • j i = (z/2)(1, 1, −1, 1) from (2.3) (items (IIIa,IIIb)). These three cases are as follows: (I) with n i ∈ Z, i = 1, . . ., 4 and A arbitrary. The above implies either (Ia) or (Ib): (Ia)ᾱ = 1 2 (a) 2 ,β = − 1 2 (a + n) 2 andγ = ǫm where m + n is even and equal to 2(n 1 − n 3 ) and a = A/2 + n 3 − n 4 arbitrary, where m + n is even and equal to 2(n 2 − n 4 ) and b = A/2 + n 1 − n 2 arbitrary (II) where A is arbitrary and n, m are integers. (IIIa) with A arbitrary and Z + that includes positive integers and zero. Accordingly, eliminating the arbitrary number A from the above equations we can writē where n = n 4 , m = n 4 + 2n 2 ∈ Z + and with n + m being an even integer. (IIIb) with A arbitrary. Z + includes positive integers and zero. Accordingly, eliminating the arbitrary number A from the above equations we can writē where n = n 2 , m = n 2 + 2n 4 ∈ Z + and with n + m being an even integer.
Comments : Integers n, m in (IIIa) and (IIIb) have been derived as positive integers. However they both enter quadratic expressions in which their overall sign can be reversed.

Applying the shift operators to obtain rational solutions
For N = 4 we will show how to reproduce items (I)-(III) listed on the page 13 in the setting of Painlevé V equation using the following construction : • The seeds of all rational solutions are the first-order polynomial solutions (2.2), (2.3) and its π variants. Note that these seeds solutions all depend on an arbitrary real parameter customary chosen here as a.
• A class of rational solutions that can be obtained by successive operation by shift operators T i , defined in the next subsection 4.3, of the form : having this singularity removed by s 1 Bäcklund transformation. These two cases are described by the parameters presented in the above items I and II, respectively.
• A class of rational solutions obtained from the seeds polynomial solutions (2.3) will be derived by successive operation with shift operators T i of the type for distinct i, j, k and Z + that contains positive integers and zero as only actions with shift operators given in equation (4.2) that are not causing divergencies. The results are summarized in the item III on page 13.
We conclude that the well-known fundamental results on classification of rational solutions of Painlevé V equation first presented in [9] are here obtained by acting with the operators (4.1) on the first-order polynomial solutions (2.2) and (2.3). In the latter case we will encounter restrictions on those values of n i for which the operators (4.1) are well-defined, as indicated in equation (4.2). See also [10] that derived rational solutions described above in items (Ia,Ib) and (II) via shift operators acting on solutions expressed by τ functions and corresponding to (2.2). The results of reference [9] were summarized succinctly in [1].

The fundamental shift operators for
To analyze transformations under the shift operators which we will introduce in this subsection it is convenient to first introduce the following representation of α i parameters for N = 4 case : One checks that is satisfied automatically without imposing any condition on v's.
Obviously adding a constant term to all v i will not change the final result in (4.3) and thus we have an equivalence: The Bäcklund transformations s i , i = 1, 2, 3 act in terms of v i simply as permutations between v i and v i+1 : The automorphism π acts as follows: π(v i ) = v i−1 , i = 2, 3, 4 and π(v 1 ) = v 4 − 1.
Next we introduce the shift operators that act as simple translations on the v i variables: T i (v j ) = v j − δ i,j leading to: .
. We are then able to associate a rational solution to each point of the orbit following approach of subsection 4.2.
It is easy to extend the definition of the fundamental shift operators to arbitrary N [16, 10, 4] : that for every N generate the weight lattice of A N −1 . The shift operators commute with each other and satisfy T 1 T 2 · · · T N = 1, where we used that π N = 1 and that πs i = s i+1 π. These operators act on parameters α i as (4.12) Within the framework of dressing chain equations with Bäcklund transformations (1.6) it is actually possible to establish a general transformation rules for the shift operator T i acting on j i+1 , j i+2 , . . . for i = 1., . . ., N , which applies to N = 4, 6 and the initial configurations (2.2), (2.4) : etc., where j i+k, n = T n i (j i+k, 0 ) with j i+k, 0 = z/N and k = 1, 2. . .. The above equations lead to (4.14) which for i = 1 will lead to recurrence relations for p = j 2 + j 3 in case of N = 4 and for p 1 = j 2 + j 3 in case of N = 6. These recurrence relations will establish Umemura polynomial solutions as will be shown below.  , 1 − a, a, 1 − a) with an arbitrary parameter a and q = p = z/2. According to relation (4.7) these solutions under action of (4.1) will have the following final parameters α 1 , α 2 , α 3 , α 4 :

Shift operators acting on the solution
Thus in agreement with item I on page 13 we find where we introduced In terms of these parameters we can decompose T n 1 1 T n 2 2 T n 3 3 T n 4 4 into a product of different factors  , 1 − a, a, 1 − a) induces the following transformations: 1. (T 1 T 3 ) n 3 increases arbitrary parameter a : a → a + 2n 3 but leaves q = p = z/2 of equation (3.5) unchanged.

(T
The conclusion in point 1. follows easily from the transformation rule : where j i = z/4 is one of the components of solution (2.2). Similar argument applies to point 2. since T 1 T 2 T 3 T 4 = 1. The first two top transformations in points, 1. and 2., do not induce any change in 1 2 (α 2 − α 4 ) nor in 1 2 (α 1 − α 3 ), thus the shift operators (T 1 T 3 ) n 3 and (T 2 T 4 ) n 4 increase equally Painlevé V parametersᾱ andβ and are not changing ǫγ parameter. The above discussion shows that the two seed configurations (a, 1 − a, a, 1 − a) and (b, 1 − b, b, 1 − b) both corresponding to the solution (3.5) with parameters a and b such that b = a + 2m, with m being an integer, can be connected by the transformation (T 1 T 3 ) n 3 (T 2 T 4 ) n 4 with m = n 3 − n 4 , that leave q = p = z/2 of equation (3.5) unchanged. Thus they both can give rise to identical solution y, (α, β, γ, δ) of the Painlevé V equation via actions of different fundamental shift operators. However this ambiguity disappears when the two seed solutions are considered as solutions (2.2) of the dressing chain since their j i (z) components will transform non-trivially under (T 1 T 3 ) n 3 (T 2 T 4 ) n 4 according to relation (4.19) as long as n 3 = n 4 .
The shift operator (T 1 T 2 ) k + increases ǫγ by 2k + , while (T 1 T −1 2 ) k − changes a difference betweenᾱ andβ of Painlevé V parameters. To illustrate how the Painlevé V parameters α,β,γ transform under the above combinations of shift operators we recall expressions (3.21) and take into account expressions (4.15) to obtain : α = (a/2 + n 3 − n 4 ) 2 2 ,β = − (a/2 + n 1 − n 2 ) 2 2 ,γ = ǫ(n 2 − n 3 − n 4 + n 1 ) . (4.20) In terms of integers k ± the above expressions can be rewritten succinctly as: Sometimes one encounters a pole in an initial expression for p like it was the case in solution (3.39), where s 1 was used to remove the pole from p. To cover such case we apply s 1 Bäcklund transformation to obtain a configuration (−a, 1, a, 1). Then applying π automorphism we arrive at (1, −a, 1, a) .
Acting with T n 1 1 T n 2 2 T n 3 3 T n 4 4 from (4.1) will yield: ,γ = ǫ(−a + n 2 − n 3 − n 4 + n 1 ), setting a = A, n 3 = 0, n 4 = −m, n 1 = n, n 2 = 0 we get item (II) on page 13, in agreement with [9], see also [10]. and p = z/2 − 9/z that contains a pole that can be removed by s 1 . Fitting the above α's into relation (4.15) does not work since the method works for p being expandable in a positive series in z. We therefore try to fit it into a structure obtained from T i 's acting on configuration (−a, 1, a, 1): it is now easy to find a class of solutions with n 3 , n 4 being arbitrary integers. If we set f.i. n 3 = n 4 = 1, then n 2 = 0 and a = −5/2 form the solution.
with a general solution given in terms of arbitrary n 3 , n 4 : that involves action by the shift operators equal to The above expression shows that there is no ambiguity related to the choice of n 3 and n 4 as (T 1 T 3 ) −1+n 3 and (T 2 T 4 ) 1+n 4 do not change the form of the solution. Therefore for simplicity we eliminate the first two factors of the above expression by choosing : and thus the action of shift operators (4.1) becomes that of T 3 T −1 4 . The action of the inverse operator T −1 4 = π −1 s 1 s 2 s 3 on p = q = z/2, (a, 1 − a, a, 1 − a) is well defined and yields Applying T 3 on the above expressions we get : 1 − a, 4 + a, −1 − a) , which for a = − 13 2 reproduces expression (3.40). 1, 1, 1) seed solution through action of the shift operators As follows from relations (4.13) applied to N = 4 case we have the following recurrence relations

Umemura polynomial solutions obtained from
T 3 (j 2 n + j 3, n ) = j 1 n + j 2, n + (1 + 2n) j 3, n + j 4, n (j 3, n + j 4, n )(j 4, n + j 1, n ) + a + 2n = j 1 n + j 2, n + (1 + 2n) z − (j 1, n + j 2, n ) (z − j 1, n − j 2, n )(z − (j 2, n + j 3, n )) + a + 2n , which can be rewritten as where for q = j 1 + j 2 , p = j 2 + j 3 we introduced the following notation Similarly from equations (4.13) we find that can be rewritten as and together with equation (4.25) form two recurrence relations for the canonical quantities q n , p n . One finds from relations (4.25) and (4.28) that which shows that the quantity d n is useful in describing transition from p n , q n to p n+1 , q n+1 . Indeed we will be able below to formulate the recurrence relation for Umemura polynomials based on existence of alternative expressions (4.34) for d n .

(4.34)
Comparing the bottom of expressions (4.33) with the two expressions in equation (4.34) we obtain two alternative recurrence relations for the Umemura polynomials which independently can be used to generate higher level Umemura polynomials. It is convenient at this point to introduce the variable x = z 2 4 and polynomials W n (x; a) = 2 −n(n−1) U n (z; a) , (4.35) which satisfy two recurrence relations that follow from comparing expressions (4.33) with (4.34) : + 60 x 4 a + 20 x 3 a 3 + 120 x 3 a 2 + 190 x 3 a + 15 x 2 a 4 + 120 x 2 a 3 + 300 x 2 a 2 + 60 x a 4 + 210 x a 3 + 6 x a 5 + 124 a 2 + 120 a 3 + 12 a 5 + a 6 + 55 a 4 , (4.41) from which higher polynomials can be obtained using recurrence relations (4.36) or (4.37). In addition, the polynomials W n (x; a) satisfy the identity 2W n+1 (x, a)W n (x; a + 1) − W n+1 (x, a + 1)W n (x; a) = W n+1 (x; a − 1)W n (x; a + 2) , (4.42) established on basis of consistency of the shift operator approach with various operators T i connected via π. Although we have chosen arbitrarily to generate the recurrence relations by acting with T 3 we could taken any other shift operator as a starting point and be able to transfer from one formalism to another by applying the automorphism π through relation πT i = T i+1 π. The identity (4.42) ensures that acting with any of the shift operators T i , i = 1, 2, 3, 4 on expressions (4.32) will give rise to solutions that are still expressible in terms of Umemura polynomials U n (z; a). For example the repeating action of T 1 operator on expressions (4.32) yields : Consider again equation (4.33) for q n (z; a) and plug q n into expression y = (q/z)(q/z − 1) −1 for solution of Painlevé V equation derived in subsection 3.2. After some simple algebra we find : .
Using the identity (4.42) to rewrite the denominator we obtain for (a, 1 − a − 2n, a + 2n, 1 − a) with the Painlevé parameters: agreeing with the solution (Ib) given at the beginning of section 4. Consider now solution (4.43a) generated by acting n times with the shift operator T 1 . The parameters α i for this solution are equal to (a + 2n, 1 − a, a, 1 − a − 2n). Plugging the above q(z) into expression y = (q/z)(q/z − 1) −1 and using the identity (4.42) we get with the Painlevé V parameters that agree with the solution (Ia) given at the beginning of section 4 for the Painlevé V variable t = 2x. The fact that the above y satisfies the Painlevé V equation is equivalent to the Umemura polynomials W n (x, a) satisfying the σ-type of relation, which can be given a form of a Toda like equation: Next we define quantity: where we suppressed dependence on n on the left hand side. It is interesting to notice that as follows from applications of all three identities (4.36), (4.37) and (4.42) ω a satisfies a discrete Painlevé II equation [11]: (4.47) See [17] for an early observation that Bäcklund transformations of continuous models can give rise to a discrete structure.

Action of the shift operators on
By acting with T n 1 1 T n 2 2 T n 3 3 T n 4 4 on j i = (z/2)(1, 1, −1, 1) from equation (2.3) with α i = (a, 0, 0, 2 − a) we will arrive, in principle, at the following parameters of the final configuration However not all of the shift transformations are well defined when acting on j i = (z/2)(1, 1, −1, 1). Since j 2 + j 3 = 0 and j 3 + j 4 = 0 we see from the definition (1.6) that actions of s 2 , s 3 involve divisions by zero and therefore are not allowed. Recalling the definitions (4.5) and (4.10) we accordingly need to exclude T 2 , T 3 and T −1 3 , T −1 4 as these operators contain s 3 and s 2 transformations at the positions to the right. Because the shift operators in (4.5) and (4.10) contain ordered products of neighboring Bäcklund transformations of the type s i+1 s i the divergence is only generated by the s i located to the right. If the result of acting by s i is not divergent then acting with s i+1 would not be divergent as follows from the definition (1.6).
Accordingly, to avoid divergencies we will only consider the operators T n 1 1 T n 4 4 T −n 2 2 with n 2 , n 4 ∈ Z + and n 1 ∈ Z.
Indeed one can verify that T −1 2 = s 2 s 3 π −1 s 1 is permissible and generates where R n (a; z) is found to satisfy the recurrence relation: R n+1 (a; z) = 2nz 2 R n−1 (a; z) + (−z 2 + 2n + a)R n (a; z), n = 1, 2, . . . , with R 0 (a; z) = 1. The solution to this recurrence relation is given by where we used the Pochhammer k-symbol (x) n,k defined as (x) n,k = x(x+2)(x+2k) · · · (x+ (n − 1)k). We notice that R n (a; z) can be expressed as a function of x = −z 2 /2 and in terms of x it holds that dR n (a; x)/dx = 2nR n−1 (a; x). Thus we find that p n from equation (4.49) satisfies p n /z = f 2 /z = d(ln R n (a; x))/dx. Based on discussion around equation (3.26) from subsection 3.3 we expect that R n (a; x) is related to Kummer's polynomial U (−n, 1 − n − a/2, x). Indeed an explicit calculation of expression (4.50) yields R n (a; x) = 2 n x n+a/2 U ( a 2 , a 2 + n + 1, − z 2 2 ), which according to relation (3.26) is equal (up to an overall constant) to U (−n, 1 − n − a/2, x), a solution to the Kummer's equation (3.24) with a = α 2 /2 = −n, b = (α 2 + α 4 )/2 = −n + 1 − a/2. Here we obtained this solution through acting n-th times with T −1 2 on the first-order solution (2.3). Since the Kummer's functions found many applications in e.g. solvable quantum mechanics, atomic physics and critical phenomena among other fields the fact that as shown above their form can be reproduced by action of the shift operators should be of potential interest for these applications and efforts to expand them.
The shift operator T 1 essentially acts as an identity its only action is to increase a → a + 2.
Let us now take a closer look at the action of T 4 on q = z, p = 0. Acting once with T 4 yields : Acting n times with T 4 on q 0 = z, p = 0 we get q n = T n 4 (q 0 ) that satisfies the recurrence relation q n = z − 2nz zq n−1 + 2n − a , (a, 0, −2n, 2(n + 1) − a) , (4.52) the corresponding expression for p n is where the zero on the right hand side follows from the recurrence relation (4.52) connecting q n , q n−1 . It we assume that F n−1 = q n−1 /z satisfies the Riccati equation (3.22) for i = 1 and α 3 = −2(n − 1) then it follows that F n = q n /z with q n determined through the recurrence relation (4.52) will satisfy the same Riccati equation (3.22) for α 3 = −2n. Since for q 0 = z the function F 0 = q 0 /z = 1 satisfies the Riccati equation (3.22) for α 3 = 0 this concludes the induction proof for q n being equal to zF a,α 3 =−2n where F a,α 3 is given by expression (3.23) in terms of Whittaker functions.
We choose a = 9, n 1 = n 2 = 0, n 3 = 2 to get the desired result. One can show for the corresponding combination of shift operators that T −2 3 = π 2 s 1 s 2 s 3 s 4 s 1 s 2 and acting with such operator on p = z, q = z and α i = (−7, 9, 0, 0) one reproduces easily the solution (4.60). Alternatively, we can obtain this solution as a special function solution when we recognize that for the condition α 4 = 0 from equation (4.60) the Hamilton equations (3.3) are solved by p = z, which when inserted in the first equation in (3.3
By comparing with results in [9] we conclude that acting with shift operators on the first-order polynomial solutions of N = 4 dressing chain produces all rational solutions of the associated Painlevé system. We therefore conjecture that the same technique will produce all rational solutions for higher even N and discuss realization of this statement for N = 6 in the next section. In this subsection we will carry out a similar discussion for the N = 6 case investigating conditions for presence of the special function solutions to the Hamilton equations (3.11). The Hamilton equations (3.11) represent four coupled nonlinear third-order differential equations. Setting to zero various components of α i introduces connections between q i , p i , i = 1, 2 and accordingly reduces a number of coupled nonlinear equations. Imposing three constraints on parameters of N = 6 Hamilton system (3.11) reduces the system to only one solvable second-order Riccati equation with a special function solution. The three constraints emerge when the two of j i are negative as in solutions (2.5)-(2.8).
When the reduced systems are realized on orbits of shift operators T n i i acting on seeds solutions (2.5)-(2.8) all these Riccati solutions become rational solutions parameterized by integers n i .

One-constraint reductions of N = 6 Hamilton Equations
We will proceed by listing possible conditions on α i parameters together with expressions for those q i , p i , i = 1, 2 that solve the reduced equations (3.11) obtained as a result of imposing constraints. For examples the formula : means that inserting the condition α 6 = 0 into the last two equations for p 1 , p 2 in (3.11) causes each of them to reduce to one identical equation for p 1 : with p 2 = z − p 1 . The reduced system of the remaining three Hamilton equations only depends on three variables q 1 , q 2 , p 1 after imposition of one single constraint. We list below other single constraints and the corresponding simple solutions for quantities entering equations (3.11) :

Multi-constraint reductions of N = 6 Hamilton Equations
One can combine the above single constraints of α i parameters into a set of two and more constraints. As we will see below the set of three constraints leads to the constrained system described by a single Riccati equation. Imposing two constraints leads as a rule to two coupled nonlinear equations but not always equations that are quadratic in underlying variables.
Let us first consider the following example of two constraints: that combines p 1 + p 2 = z that follows from α 6 = 0 and relation q 2 = z that follows from α 5 = 0. Imposing these two relations we can rewrite the Hamiltonian equations only in terms of e.g. p 2 , q 1 entering cubic non-linear equations : For the two constraints: the remaining variables p 1 , q 2 enter two coupled second-order equations : Only the first equation is a Riccati equation solvable in terms of Kummer/Whittaker functions. Next consider the two constraints The two remaining equations for q 1 , p 1 are found to be The second equation among equations (5.12) is a regular Riccati equation but the first one is a coupled Riccati equation. We will see below in Example 5.3 that the coupled equations (5.10) and (5.12) become fully solvable on orbits of the shift operators. Combining together conditions into three conditions yields one single second-order Riccati equation emerging from such reduction process.
In this case there only remains one Riccati equation for the remaining variable q 1 : (5.14) Replacing α 2 with α 4 in (5.13) yields with a Riccati equation for q 1 Similarly the three constraints leave only one Riccati equation for q 2 : zq 2, z = z(z−q 2 ) 2 +z 2 (z−q 2 )+z(α 1 +α 3 )+q 2 (1−α 1 ). Plugging f 3 = q 2 − z we get a simple looking Riccati equation for f 3 : A similar case is that of three constraints with α 3 replaced by α 1 : which leaves only one Riccati equation for q 1 : Further we also list the three constraints: As seen before α 6 = 0 leads to p 2 = z − p 1 and α 5 = 0 leads to q 2 = z. One of the remaining Hamilton equations is zq 1, z = q 1 (z − q 1 )(2p 1 − z) + zα 1 + q 1 (1 − α 1 ) with the solution q 1 = z, which when inserted in equation for p 1 gives Riccati equation : Another example of three independent constraints: The N = 6 Hamilton equations (3.11) give then for the remaining quantities q 1 , q 2 : Taking the difference of the above two equations yields equation for q 2 − q 1 which is solved for q 2 = q 1 . Thus we are left with one Riccati equation for q 1 : Another case of three constraints lead to one single Riccati equation for the remaining quantity p 1 : As seen above the three constraints reduce the four Hamiltonian equations in (3.11) to one Riccati equation for the remaining variable. As expected imposing all four constraints applied on the four Hamiltonian equations in (3.11) leads only to trivial solutions: (5.24) As we will see below in example 5.3 there are cases of two constraints with two remaining Riccati equations that decouple under special circumstances when the parameters are chosen to coincide with the orbits of the shift operators. In this subsection we will apply the fundamental shift operator techniques to I) The seed solution (2.4) with all components j i = z/N = z/6 II) The seed solution (2.5) with one of the components being negative and equal to −z/(N − 2) = −z/4 The case I) will require new class of Umemura polynomials (5.25) with the leading order term being z n(n−p) , p = 1, n/2 − 1, n/2 with the last two cases being new. In case II) we will be able to essentially reduce the problem to that of N = 4 and express the solutions in terms of regular Umemura polynomials with the leading order term being z n(n−1) . Case I). Recall the relevant N = 6 shift operators from definitions (4.11) and (4.12). For solution (2.4) with all j i = z/6, i = 1, 2, 3, 4, 5, 6 it holds that j i + j i+1 = 0 for all i = 1, 2, 3, 4, 5, 6. Thus all s i transformations acting via relation (1.6) are well defined and action by T n 1 1 T n 2 2 T n 3 3 T n 4 4 T n 5 5 T n 6 6 , n i ∈ Z, i = 1, 2, 3, 4, 5, 6 , produces rational solutions with the transformedᾱ i : (a+2n 1 −2n 2 , 2 3 −a+2n 2 −2n 3 , a+2n 3 −2n 4 , 2 3 −a+2n 4 −2n 5 , a+2n 5 −2n 6 , 2 3 −a+2n 6 −2n 1 ) .
Quite similar behavior will take place for an orbit T −n

Summary and Comments
We identified rational solutions of the dressing chain equations of even periodicity with points of an orbit generated by the fundamental shift operators acting on all first-order polynomial solutions. It was described how additional Bäcklund transformation was needed to regularize those solutions that initially contained a simple pole. For those first-order polynomial solutions which contain neighboring j i and j i+1 such that : j i + j i+1 = 0 for some 1 ≤ i ≤ N the action of some shift operators is not welldefined. Accordingly those shift operators needed to be excluded in such cases and we have described the exclusion procedure in the paper. For orbits of the remaining well-defined shift operators we showed how this structure for N = 4 is responsible for a separate class of corresponding rational solutions (item III on page 13) of Painlevé V equation. We also showed how the rational solutions generated by a single shift operator T n i are expressed by Kummer/Whittaker polynomials with arguments depending on integer n.
The advantage of the formalism we presented is that it is universal, meaning that the derivation applies to all even-cyclic dressing chain systems or equivalent A It is interesting to compare the derivation of elementary seed solutions for even-cyclic dressing chains with those encountered for odd-cyclic dressing chains. There are fundamental differences as the α i parameters are fixed and do not depend on arbitrary variables. Also in contrast to the even-cyclic dressing chains, the fundamental variables j i , i = 1, ..., N of the odd-cyclic dressing chains that satisfy equations (1.2) and the Painlevé variables f i , i = 1, ..., N are fully equivalent as the relation f i = j i + j i+1 is reversible through expression j i = 1 2 N −1 k=0 f i+k for N odd. For example for N = 3 one finds two elementary seeds solutions that can be written as j i = (z/2)(1, 1, 1), α i = (1, 1, 1) and j i = (3z)/2)(−1, 1, 1), α i = 3(0, 1, 0). It is well known that the rational solutions of the Painlevé IV equation can all be obtained by Bäcklund transformations from the above two seed solutions [18] whether expressee in terms of J i or f i .
The natural next step, which we plan to pursue in the future, is to apply this framework to obtain closed determinant or special function expressions for rational solutions of all dressing chain equations of even periodicity generated by combined shift operators.