On the equivalence of singular point quantities and the integrability of a fine critical singular point

In this paper, the equivalence of singular point quantities and the integrability of a fine critical singular point are discussed. For an extensive class of complex autonomous differential systems, the theoretical basis for the methods of using formal series and integrating factors to calculate the singular point quantities for the origin (a fine critical singular point) and judge its integrability is obtained. Two recursion formulas, which are much simpler than focal values for computation because the operation is rational, for computing singular point quantities are introduced.