We give a geometric formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and nonuniqueness of solutions of these equations, as well as their integrability.
Echeverría-Enríquez, A., Muñoz-Lecanda, M. C., & Román-Roy, N. (2002). A geometrical analysis of the field equations in field theory. International Journal of Mathematics and Mathematical Sciences, 29(12), 687–699. https://doi.org/10.1155/S0161171202013212