The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

We carry on the approach used in L. Schiavone, The inverse problem within free Electrodynamics and the coisotropic embedding theorem, Inter. J. Geom. Meth. Mod. Phys. 21(7) (2024) 2450131 to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be formulated in terms of vector fields over pre-symplectic manifolds.