symmetry - Are symmetries in the equation necessarily symmetries in the corresponding solution(s)?

I wonder whether the symmetries in the equations (such as the heat equation, the wave equation, the Schrödinger equation, Maxwell equations) are reflected into their solution(s). I.e., assuming that for a particular equation, a particular initial and boundary conditions, if the equation has, say, a rotational, translational as well as a Lorentz invariance, can I always expect the solution (assuming it's unique) to preserve these symmetries?

In electro and magnetostatics, when the equation to be solved had a rotational symmetry around, say, the z-axis, then we assumed that the solution had no dependence on the azimuthal angle coordinate. So I think that in that case, by looking at the solution, I can guess that the equation has rotational invariance around an axis and that the boundary condition also have that symmetry. Are there examples where such a supposition regarding the form of the solution fail? In other words, can a solution to an equation not contain a symmetry that the underlying equation has? If so, how can one explain it?