Non-Integrable Sub-Riemannian Geodesic Flow on $J^2(\mathbb{R}^2,\mathbb{R})$

The space of $2$-jets of a real function of two real variables, denoted by $J^2(\mathbb{R}^2,\mathbb{R})$, admits the structure of a metabelian Carnot group, so $J^2(\mathbb{R}^2,\mathbb{R})$ has a normal abelian sub-group $\mathbb{A}$. As any sub-Riemannian manifold, $J^2(\mathbb{R}^2,\mathbb{R})$ has an associated Hamiltonian geodesic flow. The Hamiltonian action of $\mathbb{A}$ on $T^*J^2(\mathbb{R}^2,\mathbb{R})$ yields the reduced Hamiltonian $H_{\mu}$ on $T^*\mathcal{H} \simeq T^*(J^2(\mathbb{R}^2,\mathbb{R})/\mathbb{A})$, where $H_{\mu}$ is a two-dimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian $H_{\mu}$ is non-integrable by meromorphic functions for some values of $\mu$. This result suggests the sub-Riemannian geodesic flow on $J^{2}(\mathbb{R}^2,\mathbb{R})$ is not meromorphically integrable.