  For $\alpha\in(0,\pi/2)$, let $(\Sigma)_\alpha$ be the control system
$\dot{x}=(F+uG)x$, where $x$ belongs to the two-dimensional unit sphere $S^2$,
$u\in [-1,1]$ and $F,G$ are $3\times3$ skew-symmetric matrices generating
rotations with perpendicular axes of respective length $\cos(\alpha)$ and
$\sin(\alpha)$. In this paper, we study the time optimal synthesis (TOS) from
the north pole $(0,0,1)^T$ associated to $(\Sigma)_\alpha$, as the parameter
$\alpha$ tends to zero. We first prove that the TOS is characterized by a
``two-snakes'' configuration on the whole $S^2$, except for a neighborhood
$U_\alpha$ of the south pole $(0,0,-1)^T$ of diameter at most $\O(\alpha)$.
  We next show that, inside $U_\alpha$, the TOS depends on the relationship
between $r(\alpha):=\pi/2\alpha-[\pi/2\alpha]$ and$\alpha$. More precisely, we
characterize three main relationships, by considering sequences
$(\alpha_k)_{k\geq 0}$ satisfying $(a)$$r(\alpha_k)=\bar{r}$; $(b)$
$r(\alpha_k)=C\alpha_k$ and $(c)$ $r(\alpha_k)=0$, where $\bar{r}\in (0,1)$ and
$C>0$. In each case, we describe the TOS and provide, after a suitable
rescaling, the limiting behavior, as $\alpha$ tends to zero, of the
corresponding TOS inside $U_\alpha$.
