Extremal trajectories in the sub-Lorentzian problem on the Engel group

Let $\mathbb{E}$ be the Engel group and let $D$ be a rank-two left-invariant distribution with Lorentzian metric on $\mathbb{E}$. The sub-Lorentzian problem is stated as the problem of maximizing the sub-Lorentzian distance. A parametrization of timelike and spacelike normal extremal trajectories is obtained in terms of Jacobi elliptic functions. Discrete symmetry groups are described in the cases of timelike and spacelike trajectories; in both cases the fixed points and the corresponding Maxwell points are calculated for each symmetry. These calculations underlie estimates for the cut time (when the trajectory ceases to be globally optimal).
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