On the density of parameterizations of generalized Fermat equations of signature (2,3,3) that produce locally primitive solutions

We consider the equations Ax^2+By^3=Cz^3, where A,B,C are square-free and pairwise co-prime integers. A solution (x,y,z) is called primitive if it consists of co-prime integers. Adapting earlier work for the equations x^2+y^3=Cz^3, we show that primitive solutions give rise to integer Klein forms of degree four, with discriminant A^3B^2C . Whether Klein forms come from primitive solutions is determined by local conditions. We show that for primes p dividing B, there are exactly four GL_2(Q_p)-equivalence classes of Klein forms that are relevant, and that exactly half of those classes come from Z_p-primitive solutions. We also show that if we set A=1, then further restricting B,C to square-free and co-prime integers leaves us with an asymptotically positive proportion of triples.