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We examine $3D$ flows $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ admitting vectoridentity $M\mathbf{v} = \nabla \times \mathbf{A}$ for a multiplier $M$ and apotential field $\mathbf{A}$. It is established that, for those systems, onecan complete the vector field $\mathbf{v}$ into a basis fitting an$\mathfrak{sl}(2)$-algebra. Accordingly, in terms of covariant quantities, thestructure equations determine a set of equations in Maurer-Cartan form. Thisrealization permits one to obtain the potential field as well as to investigatethe (bi-)Hamiltonian character of the system. The latter occurs if the systemhas a time-independent first integral. In order to exhibit the theoreticalresults on some concrete cases, three examples are provided, namely the Gulliotsystem, a system with a non-integrable potential, and the Darboux-Halphensystem in symmetric polynomials.

$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations