Let be a holomorphic vector field degenerating at such that Jacobi matrix has zero eigenvalues. Consider and let be a formal power series solution. We study the Borel summability of , which implies the existence of a genuine solution such that as in some sectorial region. Further we treat singular equations appearing in finding normal forms of singular vector fields and study to simplify by transformations with Borel summable functions.
"Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields." J. Math. Soc. Japan 57 (2) 415 - 460, April, 2005. https://doi.org/10.2969/jmsj/1158242065