A new example of finite-dimensional reduction of a discrete chain of the Toda chain type

Background. Integrable discrete equations considered in the framework of the numerical study of their continuous analogs. Same time continuous equations obtained using the limit from the discrete systems. Many respects the discrete case to be more investigate and fundamental than differential. Difference equations arise in many problems of mathematical physics. Often discrete equations studied as Bäcklund transformations of continuous or differential difference equations. The construction of finite-dimensional reductions of integrable system is one of the most effective ways to obtain their particular solutions. The aim of this paper is construction new finite-dimensional reduction of the discrete chain Toda type and analyze the integrability of the resulting reductions.
Materials and methods. The integrable equation is a consistency condition of two linear equations (L-A pair). The construction of boundary conditions and integrals of motion of finite-dimensional reductions based on these properties of an integrable discrete system. The work uses the main methods of the symmetry approach to the study of integrable systems. The methods of the theory of partial differential equations and ordinary differential equations used also.
Results. Found a new finite-dimensional reduction of a discrete chain of the Toda chain type consistent with the L-A pair. Conserved quantities and difference-differential symmetry of the resulting finite-dimensional reductions are determined. Its integrability by quadratures proved. Boundary conditions that lead the system to one of the versions of the discrete Painlevé equation $dP_{I}$ presented.
Conclusions. A simple and effective way of constructing integrable finite-dimensional reduction based on the compatibility of boundary conditions with L-A pair. Discrete analog of the Painlevé equations can be obtain a finite dimensional reductions of eh discrete Toda chains. it is necessary further study the boundary conditions compatible with the L-A pair to construct the Lax pair if the discrete Painlevé equation as a finite-dimensional reduction of an integrable chain of the Toda chain type.