Closed form of solutions of $xy-zv=±1$

Question

Let us consider a real dynamical system $s′=h(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: I am asking if this equation $$xy-zv=±1$$ has integer solutions $x,y,z,v$.

We can find some special cases, but I am interested in a closed-form.

Answer 1

By the Bezout identity, there are solutions iff $x,z$ are relative primes, and $y,v$ follow.