Particle Mechanics Models with W-symmetries

We introduce a particle mechanics model with Sp(2 M ) gauge invariance. Dif-ferent partial gauge-fixings by means of sl(2) embeddings on the gauge algebra lead to reduced models which are invariant under diffeomorphisms and classical non-linear W -transformations as the residual gauge symmetries thus providing a set of models of gauge and matter fields coupled in a W -invariant way. The equations of motion for the matter variables give Lax operators in a matrix form. We examine several examples in detail and discuss the issue of integration of infinitesimal W -transformations.