  We study gauge and gravitational field theories in which the gauge fixing
conditions are imposed as constraints on classical fields. Quantization of
fluctuations can be performed in a BRST invariant manner, while the main
novelty is that the classical equations of motion admit solutions that are not
present in the standard approach. Although the new solutions exist for both
gauge and gravitational fields, one interesting example we consider in detail
is constrained gravity endowed with a nonzero cosmological constant. This
theory, unlike General Relativity, admits two maximally symmetric solutions one
of which is a flat space, and another one is a curved-space solution of GR. We
argue that, due to BRST symmetry, the classical solutions obtained in these
theories are not ruined by quantum effects. We also comment on massive
deformations of the constrained models. For both gauge and gravity fields we
point out that the propagators of the massive quanta have soft ultraviolet
behavior and smooth transition to the massless limit. However, nonlinear
stability may require further modifications of the massive theories.
