  We prove that for each integer k of at least 2, there is an open neigborhood
\nu_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G
is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated
by elements in \nu_k, then the natural action on S^2 has non-empty fixed point
set. Moreover, the G-action has at least two fixed points if the action has a
finite non-trivial orbit.
