  We generalize K\"ahler-Ricci solitons to the almost-K\"ahler setting as the
zeros of Inoue's moment map \cite{MR4017922}, and show that their existence is
an obstruction to the existence of first-Chern-Einstein almost-K\"ahler metrics
on compact symplectic Fano manifolds. We prove deformation results of such
metrics in the $4$-dimensional case. Moreover, we study the Lie algebra of
holomorphic vector fields on $2n$-dimensional compact symplectic Fano manifolds
admitting generalized almost-K\"ahler-Ricci solitons. In particular, we
partially extend Matsushima's theorem \cite{MR0094478} to compact
first-Chern-Einstein almost-K\"ahler manifolds.
