The Generalized Fermat Equation $x^2+y^3=z^{25}$

We consider the generalized Fermat equation () $x^2+y^3=z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2+y^3=z^5$ (due to Edwards), we reduce the solution of () to the solution of five specific equations of the form $H(u,v)=w^5$, where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$, $u$ and $v$ are coprime (rational) integers, and $w\in K$.