Geometric classification of semidirect products in the maximal parabolic   subgroup of $\operatorname{Sp}(2,\mathbb{R})$
  We classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ 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. 
Filippo De Mari, Ernesto De Vito, Stefano Vigogna
math.GR