Integrable boundary conditions for the Toda lattice

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Abstract
The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding the differential constraints consistent with the ZS-AKNS hierarchy. A method of their construction is offered based on the Backlund transformations. It is shown that the generalized Toda lattices corresponding to the non-exceptional Lie algebras of finite growth can be obtained by imposing one of the four simplest integrable boundary conditions on both ends of the lattice. This fact allows, in particular, the solution of the reduction problem of the series A Toda lattices into the series D lattices. Deformations of the found boundary conditions are presented which lead to the Painleve-type equations.

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