  The partition function of the Symmetric Matrix Ensemble is identified with
the $\tau-$function of a particular solution of the Pfaff Lattice. We show that
in the case of even power interactions, in the thermodynamic limit, the
$\tau-$function corresponds to the solution of an integrable chain of
hydrodynamic type. We prove that the hydrodynamic chain so obtained is
diagonalisable and admits hydrodynamic reductions in Riemann invariants in an
arbitrary number of components.
