LINEAR R-MATRIX ALGEBRA FOR CLASSICAL SEPARABLE SYSTEMS

We consider a hierarchy of the natural-type Hamiltonian systems of n d egrees of freedom with polynomial potentials separable in general elli psoidal and general paraboloidal coordinates. We give a Lax representa tion in terms of 2 x 2 matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is p roposed. Using the method of variable separation, we provide the integ ration of the systems in classical mechanics constructing the separati on equations and, hence, the explicit form of action variables. The qu antization problem is discussed with the help of the separation variab les.