  We continue investigating the superintegrability property of matrix models,
i.e. factorization of the matrix model averages of characters. This paper
focuses on the Gaussian Hermitian example, where the role of characters is
played by the Schur functions. We find a new intriguing corollary of
superintegrability: factorization of an infinite set of correlators bilinear in
the Schur functions. More exactly, these are correlators of products of the
Schur functions and polynomials $K_\Delta$ that form a complete basis in the
space of invariant matrix polynomials. Factorization of these correlators with
a small subset of these $K_\Delta$ follow from the fact that the Schur
functions are eigenfunctions of the generalized cut-an-join operators, but the
full set of $K_\Delta$ is generated by another infinite commutative set of
operators, which we manifestly describe.
