Geometric classification of semidirect products in the maximal parabolic subgroup of $\mathrm{Sp}(2,\mathbb{R})$

We classify up to conjugation by $\mathrm{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\mathrm{Sp}(2,\mathbb{R})$ by means of an essentially geometric argument. This classification has already been established without geometry, under a stricter notion of equivalence, namely conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.