  We revisit the problem of consistent free propagation of higher-spin fields
in nontrivial backgrounds, focusing on symmetric tensor(-spinor)s. The
Fierz-Pauli equations for massive fields in flat space form an involutive
system, whose algebraic consistency owes to certain gauge identities. The zero
mass limit of the former leads directly to massless higher-spin equations in
the transverse-traceless gauge, where both the field and the gauge parameter
have their respective involutive systems and gauge identities. In nontrivial
backgrounds, it is the preservation of these gauge identities and symmetries
that ensures the correct number of propagating degrees of freedom. With this
approach we find consistent sets of equations for massive and massless
higher-spin bosons and fermions in certain gravitational/electromagnetic
backgrounds. We also present the involutive system of partially massless
fields, and give an explicit form of their gauge transformations. We consider
the Lie superalgebra of the operators on symmetric tensor(-spinor)s in flat
space, and show that in AdS space the algebra closes nonlinearly and requires a
central extension.
