  We consider the barotropic Euler equations in dimension d>1 with decaying
density at spatial infinity. The phase portrait of the nonlinear ode governing
the equation for spherically symmetric self-similar solutions has been
introduced in the pioneering work of Guderley. It allows to construct global
profiles of the self-similar problem, which however turn out to be generically
non-smooth across the associated light (acoustic) cone. In a suitable range of
barotropic laws and for a sequence of quantized speeds accumulating to a
critical value, we prove the existence of non-generic C^\infty self-similar
solutions with suitable decay at infinity. The C^\infty regularity is used in a
fundamental way in the companion papers \cite{MRRSnls}, \cite{MRRSfluid} to
control the associated linearized operator, and construct finite energy blow up
solutions of respectively the defocusing nonlinear Schr\"odinger equation in
dimension $5\le d\le9$, and the isentropic ideal compressible Euler and
Navier-Stokes equations in dimensions d=2,3.
