CHAOTIC SUBRIEMANNIAN GEODESIC FLOW IN J2(R2,R)

Both the paper and the candidate solution reduce the left-invariant geodesic Hamiltonian on T*J2(R2,R) to a 2-DOF mechanical system with quartic potential by fixing the six cyclic momenta, and then choose parameters yielding the (generalized) Yang–Mills quartic potential; they conclude non-meromorphic-integrability of the reduced system by known classifications, which implies non-integrability of the full geodesic flow. The paper states Theorem A (non-integrability) and builds the reduction via F-curves (Theorem B), then cites a classification to rule out integrability for an explicit quartic potential, thereby concluding Theorem A. The model follows the same reduction and invokes a different but standard classification (Shi–Li 2013) and the Morales–Ramis obstruction for emphasis. Minor issues in the paper include several typos and a likely mis-citation (it attributes a degree-4 classification to an item listed as degree-2). Nonetheless, the core argument and conclusion agree with the model and are correct. Key parts of the paper supporting this are Theorem A (non-integrability) and the explicit construction of the left-invariant frame, momentum functions, Hamiltonian H, and the reduction to HF with F(x,y)=F1(x)^2+F2(x,y)^2+F3(y)^2; see the statement of Theorem A and its strategy, the left-invariant frame and momentum structure, the Hamiltonian H, the F-curve construction, and the proof of Theorem A that selects a6≠0 to obtain the Yang–Mills quartic potential, from which non-integrability of HF is concluded .