The function \(g\left(t\right)\) has the interpretation of a dissipative term.
We proceed by calculating the first-order dynamical invariants related to (\ref{eq:02}) with the method proposed by \citet{Bertin_2012}. In this case we define two arbitrary complex functions \(\alpha\left(t\right)\) and \(\beta\left(t\right)\). Multiplying (\ref{eq:02a}) by \(\alpha\) and (\ref{eq:02b}) by \(\beta\), building the linear combination, and isolating the total time derivative results in the expression