Supose we have a known matrix $A:\mathbb{R}^3 \mapsto \mathbb{R}^N$ where $N>3$.
Is the a way to find all solutions $\mathbf{x}$ to
\begin{align} A\mathbf{x} = \mathbf{I} \end{align}
where $|\mathbf{x}| \leq 2,\mathbf{I} \in \mathbb{Z}^N $?
The problem here is that $\mathbf{I}$ is not known, i.e. the question really is: What vectors $\mathbf{x}$ with length $\leq 2$ hits the $\mathbb{Z}^N$ integer lattice under the map $A$?