The normal sub-Riemannian geodesic flow on $E(2)$ generated by a left-invariant metric and a right-invariant distribution

We consider a sub-Riemannian problem on the three-dimensional solvable Lie group $E(2)$ endowed with a left-invariant metric and a right-invariant distribution. The problem is based on construction of a Hamilton structure for the given metric by the Pontryagin maximum principle.