Hamilton–Jacobi theory and integrability for autonomous and non-autonomous contact systems

In this paper, we study the integrability of contact Hamiltonian systems, both time-dependent and independent. order to do so, construct a Hamilton–Jacobi theory for these systems following two approaches, obtaining different equations. Compared conservative depend one additional parameter. The fact equations reflects whether are looking solutions depending on parameter or not. illustrate developed in three examples: free particle with linear external force, freely falling dissipation damped forced harmonic oscillator.