  We show that for a large class of evolutionary nonlinear and nonlocal partial
differential equations, symmetry of solutions implies very restrictive
properties of the solutions and symmetry axes. These restrictions are
formulated in terms of three principles, based on the structure of the
equations. The first principle covers equations that allow for steady solutions
and shows that any spatially symmetric solution is in fact steady with a speed
determined by the motion of the axis of symmetry at the initial time. The
second principle includes equations that admit breathers and steady waves, and
therefore is less strong: it holds that the axes of symmetry are constant in
time. The last principle is a mixed case, when the equation contains terms of
the kind from both earlier principles, and there may be different outcomes; for
a class of such equations one obtains that a spatially symmetric solution must
be constant in both time and space. We list and give examples of more than 30
well-known equations and systems in one and several dimensions satisfying these
principles; corresponding results for weak formulations of these equations may
be attained using the same techniques. Our investigation is a generalisation of
a local and one-dimensional version of the first principle from [E., Holden,
and Raynaud, 2009] to nonlocal equations, systems and higher dimensions, as
well as a study of the standing and mixed cases.
