  The paper is devoted to quadratic Poisson structures compatible with the
canonical linear Poisson structures on trivial 1-dimensional central extensions
of semisimple Lie algebras. In particular, we develop the general theory of
such structures and study related families of functions in involution. We also
show that there exists a 10-parametric family of quadratic Poisson structures
on $\gl(3)^*$ compatible with the canonical linear Poisson structure and
containing the 3-parametric family of quadratic bivectors recently introduced
by Vladimir Sokolov. The involutive family of polynomial functions related to
the corresponding Poisson pencils contains the hamiltonian of the polynomial
form of the elliptic Calogero--Moser system.
