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 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. Parametrisations for all integral
curves are exhibited.