The moser type reduction of integrable Riccati differential equations and its Lie-algebraic structure

Abstract

A given Riccati equation, as is well known, can be naturally reduced to a system of nonlinear evolution equations on an infinite-dimensional functional manifold with Cauchy-Goursat initial data. We describe the Lie algebraic reduction procedure of nonlocal type for this infinite-dimensional dynamical system upon the set of critical points of an invariant Lagrangian functional. As one of our main results, we show that the reduced dynamical system generates the completely integrable Hamiltonian flow on this submanifold with respect to the canonical symplectic structure upon it. The above also makes it possible to find effectively its finite-dimensional Lax type representation via both the well known Moser type reduction procedure and the dual momentum mapping scheme on some matrix manifold.