  In this article, we consider the generalised two-parameter Cauchy two-matrix
model and corresponding integrable lattice equation. It is shown that with
parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the
average characteristic polynomials admit $(k_1+k_2+2)$-term recurrence
relations, which provide us spectral problems for integrable lattices. The tau
function is then given by the partition function of the generalised Cauchy
two-matrix model as well as Gram determinant. The simplest example with exact
solvability is demonstrated.
