Positive integral points on Mordell-Schinzel surfaces

We apply the theory of generalised cluster algebras to classify the finiteness of positive integral points on the class of log K3 surfaces defined by $xyz = G(x,y)$ where $G$ has positive coefficients. We furthermore establish results on Zariski density, number of orbits and asymptotic growth of such points. These provide a Diophantine realisation of the finite-affine-indefinite trichotomy of rank-two root systems.