We study the construction of the intrinsic action for PDEs equipped with the compatible presymplectic structures. In particular, we explicitly demonstrate that the intrinsic action for the standard Einstein-Hilbert gravity is the familiar first-order Palatini action. We then study the example of massive spin-2 field, where the natural presymplectic structure is not complete in the sense that the associated intrinsic action does not reproduce all the equations of motion. We explicitly relate this to the differential consequences of zeroth order in the genuine Lagrangian formulation of Fierz and Pauli. Moreover, we construct a minimal multisymplectic extension of the intrinsic action that produces all the equations of motion and argue that systems of this type can be naturally regarded as multidimensional analogs of mechanical systems with constraints.
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