The simplest and commonest class of mathematical models generating spiral waves is the reaction diffusion systems on the plane,
with 
, 
,
. Our arguments can be extended to a much
wider class of systems, as the essential point lies in the symmetry.
Apart from the invariance against shifts in time, (1) is invariant under the group of the isometric
transformations of the plane 
, the
Euclidean group denoted E(2) (we shall neglect reflections and
consider only the orientation-preserving transformations). That is, if
is a solution to (1), then 
is another solution, for any 
, where
action T(g) of on the function u is defined as
We are interested in spiral wave solutions of such systems, though it is not easy to specify this class of solution formally. Rigidly rotating waves are independent of time in an appropriately rotating frame of reference; however, as we are interesting in meandering not rigidly rotating waves, this circumstance is not much helpful. In fact, for our current purposes it is enough to mention that the isotropy subgroup of spiral wave solutions is trivial, i.e. they are not invariant under any nontrivial Euclidean transformation,
It is well known that the behaviour of dynamical systems with symmetries can be drastically different from those without symmetry, i.e. generic systems (Anosov & Arnold, 1985), and a standard way to study symmetrical systems is to reduce them to generic ones and then apply the results of the generic theory. For continuous groups, this can be made by separation of the system movement onto superposition of that `along the group' and `across the group', the second being described by a generic vector field that lacks the symmetry of the parent system (1).
