I am posting this as a followup to this question in the hopes that a more specific question might elicit an answer.
Let $p$ be prime and $n>1$ an integer, and consider the system of equations $\left[\begin{array}{cc} a & b \\ c & d\end{array}\right] \cdot \left[ \begin{array}{c} x \\ y \end{array}\right] = {\bf 0} \bmod p^n$ in the unknowns $x, y$ (i.e., $a, b, c, d$ are given).
How can we determine how many nonzero solutions there are to this system of equations?
(Obviously if $n=1$ then this is all easy; I am surprised(?) that the more general case seems to be barely discussed.)