Integral Hasse principle for Markoff type cubic surfaces

We establish new upper bounds on the number of failures of the integral Hasse principle within the family of Markoff type cubic surfaces $x^2+ y^2+ z^2- xyz= a$ with $|a|\leq A$ as $A\to \infty$. Our bound improves upon existing work of Ghosh and Sarnak. As a result, we demonstrate that the integral Hasse principle holds for a density $1$ of surfaces in certain sparse sequences.