  This paper is devoted to the generic transversality of radially symmetric
stationary solutions of nonlinear parabolic systems of the form \[ \partial_t
w(x,t) = -\nabla V\bigl(w((x,t))\bigr) + \Delta_x w(x,t) \,, \] where the space
variable $x$ is multidimensional and unbounded. It is proved that, generically
with respect to the potential $V$, radially symmetric stationary solutions that
are stable at infinity (in other words, that approach a minimum point of $V$ at
infinity in space) are transverse; as a consequence, the set of such solutions
is discrete. This result can be viewed as the extension to higher space
dimensions of the generic elementarity of symmetric standing pulses, proved in
a companion paper. It justifies the generic character of the discreteness
hypothesis concerning this set of stationary solutions, made in another
companion paper devoted to the global behaviour of (time dependent) radially
symmetric solutions stable at infinity for such systems.
