On p-form vortex-lines equations on extended phase space

In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as $i_{\dot \gamma}d\sigma =0$, where $\sigma$ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where $\sigma$ is a differential p-form.