  We consider the second variational derivative of a given gauge-natural
invariant Lagrangian taken with respect to (prolongations of) vertical parts of
gauge-natural lifts of infinitesimal principal automorphisms. By requiring such
a second variational derivative to vanish, {\em via} the Second Noether Theorem
we find that a covariant strongly conserved current is canonically associated
with the deformed Lagrangian obtained by contracting Euler--Lagrange equations
of the original Lagrangian with (prolongations of) vertical parts of
gauge-natural lifts of infinitesimal principal automorphisms lying in the
kernel of the generalized gauge-natural Jacobi morphism.
