Analytic and Computer Study of the Simplest almost Riemannian Problem with a Martinet Point

The simplest almost Riemannian problem with a Martinet point is studied. Extremal and optimal trajectories are studied by analytic, symbolic, and numeric techniques.
Analytically the following results were obtained: existence of optimal trajectories was proved, absence of abnormal trajectories was shawn, a Hamiltonian system for normal extremals was derived, its symmetry and Maxwell points was described. Symbolically all polynomial integrals of the Hamiltonian system of degree not greater than 54 were described. Numerically the optimal synthesis was constructed.