  In this paper, the authors investigate the case of discrete multiple
orthogonal polynomials with two weights on the step line, which satisfy Pearson
equations. The discrete multiple orthogonal polynomials in question are
expressed in terms of tau functions, which are double Wronskians of generalized
hypergeometric series. The shifts in the spectral parameter for type II and
type I multiple orthogonal polynomials are described using banded matrices. It
is demonstrated that these polynomials offer solutions to multicomponent
integrable extensions of the nonlinear Toda equations. Additionally, the paper
characterizes extensions of the Nijhoff-Capel totally discrete Toda equations.
The hypergeometric $\tau$-functions are shown to provide solutions to these
integrable nonlinear equations. Furthermore, the authors explore Laguerre-Freud
equations, nonlinear equations for the recursion coefficients, with a
particular focus on the multiple Charlier, generalized multiple Charlier,
multiple Meixner II, and generalized multiple Meixner II cases.
