A square matrix is  called non-derogatory (or cyclic) if its (unitary) characteristic and minimal polynomials coincide. A non-derogatory matrix gives rise to a $2$-step solvable Lie algebra endowed with an exact
symplectic structure (hence a family of locally isomorphic $2$-step solvable Lie groups possessing left invariant exact symplectic structures). We classify such Lie groups via their Lie algebras.