  We study the construction of the so-called intrinsic action for PDEs equipped
with 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. Our main focus in this work is the
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 feature to the differential
consequences of the zeroth order in the genuine Lagrangian formulation of Fierz
and Pauli. Moreover, a minimal multisymplectic extension of the intrinsic
action that produces all the equations of motion is constructed and it is
argued that systems of this type can be naturally regarded as multidimensional
analogs of mechanical systems with constraints. Finally, we extend the
considerations to the massive spin-3 field and argue that the extension to all
the massive higher spins should be straightforward as well.
