Geometric classification of semidirect products in the maximal parabolic subgroup of Sp(2, ℝ)

We classify up to conjugation by GL(2, ℝ) (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of Sp(2, ℝ) by means of an essentially geometric argument. This classification has already been established in [G. S. Alberti, L. Balletti, F. De Mari and E. De Vito, Reproducing subgroups of Sp(2, ℝ). Part I: Algebraic classification, J. Fourier Anal. Appl.9(4) (2013) 651–682] 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.