Geodesic
Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater
Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 672-713

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One of the main approaches to investigating sub-Riemannian problems is Mitchell's theorem on nilpotent approximation, which reduces the analysis of a neighbourhood of a regular point to the analysis of the left-invariant sub-Riemannian problem on the corresponding Carnot group. Usually, the in-depth investigation of sub-Riemannian shortest paths is based on integrating the Hamiltonian system of Pontryagin's maximum principle explicitly. We give new formulae for sub-Riemannian geodesics on a Carnot group with growth vector $(2,3,5,6)$ and prove that left-invariant sub-Riemannian problems on free Carnot groups of step 4 or greater are Liouville nonintegrable. Bibliography: 30 titles.
Keywords: sub-Riemannian geometry, Liouville integrability, Carnot groups, growth vector, separatrix splitting, Melnikov-Poincaré method. 
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     author = {L. V. Lokutsievskiy and Yu. L. Sachkov},
     title = {Liouville integrability of {sub-Riemannian} problems on {Carnot} groups of step 4 or greater},
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     number = {5},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_5_a3/}
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L. V. Lokutsievskiy; Yu. L. Sachkov. Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater. Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 672-713. http://geodesic.mathdoc.fr/item/SM_2018_209_5_a3/