Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue

We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form $xy-zw=r$, where $r$ is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables $x, y, z, w$ as well as of $r$. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence $xy-zw \equiv 1 (\text{mod }p)$, where $p$ is a large prime, with a strong bound on the error term.