  For an exponential Lie group $G$ and an irreducible unitary representation
$(\pi,\mathcal{H}_{\pi})$ of $G$, we consider the natural action defined by
$\pi$ on the projective space of $\mathcal{H}_{\pi}$, and show that the
stabilisers of this action coincide with the projective kernel of $\pi$. Using
this, we prove that, if $G/\mathrm{pker}(\pi)$ is unimodular, then $\pi$ admits
a symplectic projective orbit if and only if $\pi$ is square-integrable modulo
its projective kernel $\mathrm{pker}(\pi)$.
