Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields

Let L = Sigma(d)(i=1) X-i (z) delta(zi) be a holomorphic vector field degenerating at z = 0 such that Jacobi matrix ((delta X-i/delta z(j))(0)) has zero eigenvalues. Consider Lu = F(z,u) and let (u) over tilde (z) be a formal power series solution. We study the Borel summability of (u) over tilde (z), which implies the existence of a genuine solution u(z) such that u(z) similar to (u) over tilde (z) as z -> 0 in some sectorial region. Further we treat singular equations appearing in finding normal forms of singular vector fields and study to simplify L by transformations with Borel summable functions.