ON A 2-DIMENSIONAL DARBOUX SYSTEM - INTEGRABLE REDUCTIONS(1)

A systematic way of obtaining integrable reductions of a classical sys tem investigated by Darboux in connection with conjugate coordinate sy stems is presented. It includes, in particular, the Lame system, its g eneralization to pseudo-Riemannian spaces of constant curvature, an in tegrable 2+1-dimensional sine-Gordon equation and a hyperbolic equatio n of Klein-Gordon type. The integrability of a classical generalized W eingarten system set down by Bianchi is proven by means of a suitable superposition of two constraints. It is shown that these reductions ar e preserved under a Darboux-Levi-type transformation. A connection to the Moutard transformation is recorded.