Invariant manifolds and separation of the variables for integrable chains

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier, it has been observed that this kind of objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that an ordinary difference equation (ODE) defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive an ODE, for the roots of the polynomials. The efficiency of the method is approved by some illustrative examples.