  In this paper we use the Riemann-Hilbert problem, with jumps supported on
appropriate curves in the complex plane, for matrix biorthogonal polynomials
and apply it to find Sylvester systems of differential equations for the
orthogonal polynomials and its second kind functions as well. For this aim,
Sylvester type differential Pearson equations for the matrix of weights are
shown to be instrumental. Several applications are given, in order of
increasing complexity. First, a general discussion of non-Abelian Hermite
biorthogonal polynomials in the real line, understood as those whose matrix of
weights is a solution of a Sylvester type Pearson equation with coefficients
first order matrix polynomials, is given. All these is applied to the
discussion of possible scenarios leading to eigenvalue problems for second
order linear differential operators with matrix eigenvalues. Nonlinear matrix
difference equations are discussed next. Firstly, for the general Hermite
situation a general non linear relation (non trivial because the non
commutativity features of the setting) for the recursion coefficients is
gotten.
  In the next case of higher difficulty, degree two polynomials are allowed in
the Pearson equation, but the discussion is simplified by considering only a
left Pearson equation. In the case, the support of the measure is on an
appropriate branch of an hyperbola. The recursion coefficients are shown to
fulfill a non-Abelian extension of the alternate discrete Painlev\'e I
equation. Finally, a discussion is given for the case of degree three
polynomials as coefficients in the left Pearson equation characterizing the
matrix of weights. However, for simplicity only odd polynomials are allowed. In
this case, a new and more general matrix extension of the discrete Painlev\'e I
equation is found.
