Let $Z = \mathbb{C}[x, y]$, the polynomial ring in two variables over $\mathbb{C}$, and $\mathfrak{m} = (x - 1, y)$. Let $a, b \in \mathfrak{m}$. Consider the following linear equation of $u, v$ in $Z_{\mathfrak{m}}$ (the localization of $Z$ at $\mathfrak{m}$): $$\begin{bmatrix} a & - bx \\-b & a\end{bmatrix} \begin{bmatrix} u \\v \end{bmatrix} = 0.$$ This linear system has nonzero solution in $Z_{\mathfrak{m}}$, since the determinant of its coefficient matrix is not invertible in $Z_{\mathfrak{m}}$. I wonder how to describe the solutions of this linear system? Is there a way to solve this system?
Thank you!