For a prime $p$ consider non-zero integers $x,y,z$ that satisfy: $$ x^2 + p y^2 = z^3$$
Does this fit in a known class of Diophantine equations that have been studied already?
I'm not sure how to go about solving these. Looking at it mod $p$, it looks like it should be easy to just choose a $z$, cube it, check if the result is a quadratic residue and solve for $x$. But I'm not sure how to lift this to a solution in the integers. Or maybe that is just a horrible starting approach.
How can I find solutions to this equation?