Further Reduction of Nilpotent Three-dimensional Vector Fields' Normal Form Using First Integral

To research hypernormal form (simplest normal form, unique normal form) of nilpotent three- dimensional vector field, the authors obtained the first normal form of this system using normal form theory of ordinary differential equation and dynamical systems, and applying new grading function and the first integral of the linear part. With further reduction the second order normal form was got under the condition that the coefficient of second-degree term was not equal to zero, and its uniqueness was proved.