Integral solutions of a Fermat near miss

In this talk I will consider a type of Diophantine equations called "Fermat near misses". Studying the existence and abundance of integral solutions of this kind of equations can be translated in studying the integral points on a del Pezzo surface of degree 2. Rational points on these surfaces have been widely studied, but results on integral points are still missing.
Using a guiding example, I will present a joint work (in progress) with Dan Loughran (University of Bath), where we show that on this surface we have "many" integral points (a not thin set).