  We consider spherically symmetric space-times in GR under the unconventional
assumptions that the spherical radius $r$ is either a constant or has a null
gradient in the $(t,x)$ subspace orthogonal to the symmetry spheres (i.e.,
$(\partial r)^2 = 0$). It is shown that solutions to the Einstein equations
with $r = \rm const$ contain an extra (fourth) spatial or temporal Killing
vector and thus satisfy the Birkhoff theorem under an additional physically
motivated condition that the lateral pressure is functionally related to the
energy density. This leads to solutions that directly generalize the
Bertotti-Robinson, Nariai and Plebanski-Hacyan solutions. Under similar
conditions, solutions with $(\partial r)^2 = 0$ but $r\ne\rm const$, supported
by an anisotropic fluid, contain a null Killing vector, which again indicates a
Birkhoff-like behavior of the system. Similar space-times supported by pure
radiation (in particular, a massless radiative scalar field) contain a null
Killing vector without additional assumptions, which leads to one more
extension of the Birkhoff theorem. Exact radial wave solutions have been found
(i) with an anisotropic fluid and (ii) with a gas of radially directed cosmic
strings (or a "string cloud") combined with pure radiation. Furthermore, it is
shown that a perfect fluid with isotropic pressure and a massive or
self-interacting scalar field cannot be sources of gravitational fields with a
null but nonzero gradient of $r$.
