Transitive subgroups of transvections acting on some symplectic symmetric spaces of Ricci type

Symmetric symplectic spaces of Ricci type are a class of symmetric symplectic spaces which can be entirely described by reduction of certain quadratic Hamiltonian systems in a symplectic vector space. We determine, in a large number of cases, whether such a space admits a subgroup of its transvection group acting simply transitively. We observe that the simply transitive subgroups obtained are one-dimensional extensions of the Heisenberg group.