This paper presents a unified framework for studying dynamics and integration on q-cosymplectic manifolds. After outlining the geometric foundations of q-cosymplectic structures, we derive new results concerning integrable systems and the characterization of Liouville coordinates, and further investigate the Lie integrability of q-evolution systems in this setting. We then develop a Hamilton--Jacobi theory tailored to multi-time Hamiltonian systems, both from an intrinsic geometric perspective and via symplectification techniques. To illustrate the applicability of the framework, we construct a q-cosymplectic Hamiltonian model for an extended FitzHugh-Nagumo system, providing a biologically relevant example involving three distinct temporal scales.
Leok et al. (Sat,) studied this question.