  The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating
nonlinear sigma models and Skyrme models is revisited and its generalization
for non-spherically symmetric spacetimes is proposed. The key idea behind our
construction is that, even if the matter fields depend on the Killing
coordinates in a nontrivial way, the corresponding energy-momentum tensor can
still be compatible with spacetime symmetries. Our generalized hedgehog ansatz
reduces the Skyrme equations to coupled differential equations for two scalar
fields together with several constraint equations between them. Some particular
field configurations satisfying those constraints are presented in several
physically important spacetimes, including stationary and axisymmetric
spacetimes. Incidentally, several new exact solutions are obtained under the
standard hedgehog ansatz, one of which represents a global monopole inside a
black hole with the Skyrme effect.
