  Any recipe to grow black hole hair has to circumvent no-hair theorems by
violating some of their assumptions. Recently discovered hairy black hole
solutions exist due to the fact that their scalar fields don't inherit the
symmetries of the spacetime metric. We present here a general analysis of the
constraints which limit the possible forms of such a hair, for both the real
and the complex scalar fields. These results can be taken as a novel piece of
the black hole uniqueness theorems or simply as a symmetry noninheriting
Ans\"atze guide. In addition we introduce new classification of the
gravitational field equations which might prove useful for various
generalizations of the theorems about spacetimes with symmetries.
