On the degeneration of Kovalevskaya exponents of Laurent series solutions of quasi-homogeneous vector fields

A structure of families of Laurent series solutions of a quasi-homogeneous vector field is studied, where a given vector field is assumed to have a commutable vector field. For an $m$ dimensional vector field, a family of Laurent series solutions is called principle if it includes $m$ arbitrary parameters, and called non-principle if the number is smaller than $m$. Starting from a principle Laurent series solutions, a systematic method to obtain a non-principle Laurent series solutions is given. In particular, from the Kovalevskaya exponents of the principle Laurent series solutions, which is one of the invariants of quasi-homogeneous vector fields, the Kovalevskaya exponents of the non-principle Laurent series solutions are obtained by using the commutable vector field.