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.
