Any two-input left-invariant control affine system of full rank, evolving on theEuclidean group SE (2), is (detached) feedback equivalent to one ofthree typical cases. In each case, we consider an optimal control problem which is thenlifted, via the Pontryagin Maximum Principle, to a Hamiltonian system onthe dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the maintopic of this paper. A qualitative analysis of each reduced system is performed. Thisanalysis includes a study of the stability nature of all equilibrium states, as well asqualitative descriptions of all integral curves. Finally, the reduced Hamilton equationsare explicitly integrated by Jacobi elliptic functions. Parametrisations for all integralcurves are exhibited.
