Particle Mechanics Models with W-symmetries

We introduce a particle mechanics model with Sp($2M$) gauge invariance. Different 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.