Summary: Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE(2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin maximum principle, to a Hamiltonian system on the dual space \(\mathfrak{se}(2)^*\). These reduced Hamilton-Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrizations for all integral curves are exhibited.