Applying Hasse's Principle To Arbitrary Quadratic Form

Question

Given an arbitrary equation $xy = c$ with c being a known constant and the constraint $\langle x,y \vert x,y \in \Bbb Z\rangle$ , we can further write this as a representative of 0, $xy-c = 0$ and then homogenize it to $xy-cz^2 = 0$. The Hasse-Minkowski theorem holds that Hasse's principle holds for Quadratic Forms representing 0. In the case that we already know c has integer solutions without having to solve over the p-adics and rationals, how specifically would we find the integer solutions to this equation? The best I could find was "piecing integer solutions together using the Chinese Remainder Theorem". Any help is appreciated.