INTEGRABLE BOUNDARY-CONDITIONS FOR THE TODA LATTICE

The problem of construction of the boundary conditions for the Toda la ttice compatible with its higher symmetries is considered. It is demon strated that this problem is reduced to finding the differential const raints consistent with the ZS-AKNS hierarchy. A method of their constr uction is offered based on the Backlund transformations. It is shown t hat 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 latti ce. This fact allows, in particular, the solution of the reduction pro blem of the series A Toda lattices into the series D lattices. Deforma tions of the found boundary conditions are presented which lead to the Painleve-type equations.