  We consider spherical data $X_i$ noised by a random rotation
$\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$,
$i=1,\dots, N$ is observed. We define a nonparametric test procedure to
distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on
the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \C\psi_N$ and $f$ is in a Sobolev
space with smoothness $s$''. For a noise density $f_\varepsilon$ with
smoothness index $\nu$, we show that an adaptive procedure (i.e. $s$ is not
assumed to be known) cannot have a faster rate of separation than
$\psi_N^{ad}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ and we provide a
procedure which reaches this rate. We also deal with the case of super smooth
noise. We illustrate the theory by implementing our test procedure for various
kinds of noise on SO(3) and by comparing it to other procedures. Applications
to real data in astrophysics and paleomagnetism are provided.
