I look for a solver or reference rather than an answer about how to solve the diophantine matrix equation. One states it below.
$\mathbf{X}\mathbf{N} + \mathbf{Y}\mathbf{D} = \mathbf{I}$
All entries of X, N, Y and D belongs to polynomials. Therefore, it is cumbersome to achieve.
I thank in advance.
Solving that system is of course equivalent to solving $$\begin{pmatrix}X & Y\end{pmatrix} \begin{pmatrix}N \\ D\end{pmatrix} = I$$
For examples of algorithms commonly used in computer algebra systems to solve such a system over the integers, look at:
J. D. Dixon. Exact solution of linear equations using p-adic expansion. Numerische Mathematik, 40:137–141, 1982.
T.W.J. Chou and G.E. Collins., Algorithms for the solution of systems of linear Diophantine equations, SIAM J. Computing, 11(4):687--708, 1982.