I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being

$x^2=4+8y^2+13z^2$.

The ideal answer would be a way to parametrize all the integer equations. It is easy to find infinitely many solutions. For instance, by setting $y=z$, we can use the solutions of Pell's equation. Is there a more general technique to get other solutions? Note that the techniques used for the Legendre's equation do not apply since this equation is not homogeneous.