@article{Nikonorov2025, 
author = {Yuriĭ G. Nikonorov and Irina A. Zubareva},
title = {On the behavior of geodesics of left-invariant sub-Riemannian metrics on the group        Aff          0        ⁡  (      R    )  ×      Aff          0        ⁡  (      R    )},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {1},
pages = {181-209},
keywords = {extremal, generating subspace of a Lie algebra, Hamiltonian system, Kovalevskaya exponents, left-invariant sub-Riemannian metric, non-commutative two-dimensional Lie group, sub-Riemannian geodesic},
url = {https://www.sciopen.com/article/10.3934/era.2025010},
doi = {10.3934/era.2025010},
abstract = {In this paper, we study geodesics of left-invariant sub-Riemannian metrics on the Cartesian square of a connected two-dimensional non-commutative Lie group, where the metric is determined by the inner product on a two-dimensional generating subspace of the corresponding Lie algebra. It is proven that the system of equations for geodesics of such a sub-Riemannian metric is not completely integrable in the class of meromorphic functions. Important qualitative characteristics of the corresponding geodesics are found, thus proving the complexity of their behavior in general.}
}