Complete integrability classification of the reduced b=0 system

Classify the integrability of the reduced two-dimensional trigonometric Nosé–Hoover system obtained when \(b=0\), including determining whether exceptional integrable cases exist beyond the cases identified through Darboux polynomials and exponential factors.

Background

When $b=0$, the thermostat variable $z$ is constant, and the three-dimensional system reduces on each invariant torus $z=c$ to a two-dimensional system with parameter $\kappa=a\sin c$. The paper analyzes this reduced system qualitatively and applies Darboux theory, finding explicit structures for selected parameter values but not a classification of all possible integrability phenomena.

The authors explicitly state that a complete classification remains unresolved and specifically pose the question of whether additional exceptional integrable cases occur beyond those already identified.