  An integrable system is often formulated as a flat connection, satisfying a
Lax equation. It is given in terms of compatible systems having a common
solution called the ``wave function" $\Psi$ living in a Lie group $G$, which
satisfies some differential equations with rational coefficients. From this
wave function, it is usual to define a sequence of ``correlators" $W_n$, that
play an important role in many applications in mathematical physics. Here, we
show how to systematically obtain ordinary differential equations (ODE) with
polynomial coefficients for the correlators. An application is random matrix
theory, where the wave functions are the expectation value of the
characteristic polynomial, they form a family of orthogonal polynomials, and
are known to satisfy an integrable system. The correlators are then the
correlation functions of resolvents or of eigenvalue densities. We give the ODE
and recursion on the matrix size that they satisfy.
