  We consider the free propagation of totally symmetric massive bosonic fields
in nontrivial backgrounds. The mutual compatibility of the dynamical equations
and constraints in flat space amounts to the existence of an Abelian algebra
formed by the d'Alembertian, divergence and trace operators. The latter, along
with the symmetrized gradient, symmetrized metric and spin operators, actually
generate a bigger non-Abelian algebra, which we refer to as the "consistency"
algebra. We argue that in nontrivial backgrounds, it is some deformed version
of this algebra that governs the consistency of the system. This can be
motivated, for example, from the theory of charged open strings in a background
gauge field, where the Virasoro algebra ensures consistent propagation. For a
gravitational background, we outline a systematic procedure of deforming the
generators of the consistency algebra in order that their commutators close. We
find that equal-radii AdSp X Sq manifolds, for arbitrary p and q, admit
consistent propagation of massive and massless fields, with deformations that
include no higher-derivative terms but are non-analytic in the curvature. We
argue that analyticity of the deformations for a generic manifold may call for
the inclusion of mixed-symmetry tensor fields like in String Theory.
