I have a question regarding the Weierstrass-type cubic equation

$$ Y^2 = 4 X^3 - g_2(z) X - g_3(z), $$

where the functions $g_2(z)$ and $g_3(z)$ belong to some function field $\mathcal{F}$ or ring $\mathcal{R}$. For example, $\mathcal{F}$ can be the field $\mathbb{C}(z)$ of rational functions and $\mathcal{R}$ the ring $\mathbb{C}[z]$ of polynomials (or, even, $\mathcal{F}$ can be the field of meromorphic functions and $\mathcal{R}$ the ring of entire functions).

Naturally, one can consider the associated Diophantine-type problem, namely to look for solutions $X = X(z)$, $Y = Y(z)$ belonging to the field $\mathcal{F}$ or to the ring $\mathcal{R}$.

Is there any literature on such problems? For example, is there a Mordell-type or a Siegel-type theorem?