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.