Equations and Symmetries in Covariant Hamiltonian Field Theories

We state the intrinsic form of the Hamiltonian equations of first-order Classical field theories using multivector fields. Then we study the ex-istence and non-uniqueness of solutions, and the integrability problem for these equations. Furthermore, we consider the existence of first inte-grals of these Hamiltonian equations, introducing the notion of Cartan-Noether symmetries and general symmetries of the system, and stating different versions of Noether’s theorem in this context: the “classical” one and its generalization to include higher-order Cartan-Noether sym-metries.