  In spherical symmetry compelling numerical evidence suggests that in general
relativity solutions near the threshold of black hole formation exhibit
critical behavior. One aspect of this is that threshold solutions themselves
are self-similar and are, in a certain sense, unique. To an extent yet to be
fully understood, the same phenomena persist beyond spherical symmetry. It is
therefore desirable to construct models that exhibit such symmetry at the
threshold of blow-up. Starting with deformations of the wave equation, we
discuss models which have discretely self-similar threshold solutions. We study
threshold solutions in the past light cone of the blow-up point. In spherical
symmetry there is a sense in which a unique critical solution exists. Spherical
numerical evolutions are also presented for more general models, and exhibit
similar behavior. Away from spherical symmetry threshold solutions attain more
freedom. Different topologies of blow-up are possible, and even locally the
critical solution needs reinterpretation as a parameterized family.
