There is a famous nonradiating configuration, which is not exactly what you asked, but it is very interesting, since it is also true in GR. If you have a sphere of fixed total charge whose radius oscillates arbitrarily, there is no electromagnetic radiation emitted. The solution is just given by Gauss's law outside the sphere, and you can verify that this satisfies Maxwell's equations even in the presence of everywhere equal radial currents, since the magnetic fields produced by these currents cancel out.
The deep reason that this doesn't radiate is that photons are spin 1, so they can't be emitted by an always spherically symmetric source distribution. The analogous Birkhoff theorem in GR tells you that a radially oscillating massive sphere doesn't radiate, although there it's because gravitons are spin 2.