  Matrix integrals used in random matrix theory for the study of eigenvalues of
matrix ensembles have been shown to provide $ \tau $-functions for several
hierarchies of integrable equations. In this paper, we construct the matrix
integral solutions to the Leznov lattice equation, semi-discrete and
full-discrete version and the Pfaffianized Leznov lattice systems,
respectively. We demonstrate that the partition function of Jacobi unitary
ensemble is a solution to the semi-discrete Leznov lattice and the partition
function of Jacobi orthogonal/symplectic ensemble gives solutions of the
Pfaffianized Leznov lattice.
