  Poincar\'e proved nonexistence of formal first integrals near a nonresonant
singularity of analytic autonomous differential systems. In the resonant case
with one zero eigenvalue and others nonresonant, there remains an open problem
on regularity and convergence of local first integrals. Here we provide an
answer to this problem. The system has always a local $C^\infty$ first integral
near the singularity when it is nonisolated. In any finite dimensional space
formed by analytic differential systems having the same linear part at the
singularity, either all the systems have local analytic first integrals or only
the systems in a pluripolar subset have local analytic first integrals.
