  We consider a hierarchy of the natural type Hamiltonian systems of $n$
degrees of freedom with polynomial potentials separable in general ellipsoidal
and general paraboloidal coordinates. We give a Lax representation in terms of
$2\times 2$ 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 proposed. Using the
method of variable separation we provide the integration of the systems in
classical mechanics conctructing the separation equations and, hence, the
explicit form of action variables. The quantisation problem is discussed with
the help of the separation variables.
