To apply the above construction to the spiral waves, we choose
and its representation T on
given by (2). The choice of
the Banach space is not quite obvious: e.g., we cannot use
since spiral waves do not vanish at infinity and thus do not belong to this
space, and we cannot use 
since arbitrary small rotations can produce finite
changes in functions of this space and so the representation T is not
differentiable. Hence, consists of bounded continuous
vector functions which are asymptotically
``circular'' at infinity, so that small rotations change them slightly; a
formal construction of such a space can be found in
(Wulff, 1996a, b).
Condition (3) means that the isotropy subgroup of spiral waves is
trivial. So, all we need is to choose the reduction manifold 
to
satisfy the transversality condition, which for our problem means that we
should define a class of functions 
by conditions
which would be violated by any motion of the plane. A simple and obvious
choice of such conditions is
with appropriately chosen constants 
and 
; components 1
and 2 are chosen just for example. The idea is that first and second
conditions make impossible translations -- small or finite, if they
originate, locally or globally, from a unique solution,
while the third one makes impossible rotations -- if gradient of 
at the origin is nonzero. For small transformations, this can be
guaranteed e.g. by 
(finite transformations will be discussed in the next section).
Generators of the representation T defined by (2) are
, 
and 
, for translations along x, y and rotation around the
origin respectively. Expanding 
in
this basis, to
brings (9) to the form
where 
can be interpreted as a
translation velocity and 
as a rotation velocity. The system of
PDE (12) and finite equations (11) can be viewed as a
dynamical system in the phase space 
where v is a vector-function of 
and 
, 
and 
are scalar variables. This is the target quotient
system, corresponding to the abstract quotient system (9).
Equation (10) for g(t) is easy to treat by using the isomorphism
between the plane 
on which the wave rotates, and the
complex plane 
. A natural representation 
of
E(2) on C is the group of similar movements of the complex
plane, i.e. if 
is rotation through an angle 
around the origin, followed by translation by vector 
,
then g is represented by
where 
. Infinitesimal transformation 
is represented in
C by 
, and in the functional space by 
. Thus Eq. (9) is
represented in C by
Substition of the definition of (13) gives
where obviously 
, which leads to
This is the equation on the group in the coordinates 
