Unique normal form of a class of 3 dimensional vector fields with symmetries

•Provide unique normal form for a 3-dimensional vector field with symmetries.•Apply linear grading function and multiple Lie Brackets to get unique normal form.•Simplify expression of block matrices involved in computation by a new notation.•Apply induction method successfully for computing huge size determinants.•Show consistency between unique normal form in degenerate case with BT-singularity.

We present the unique normal form of a class of 3 dimensional vector fields (BT-zero singularity) with symmetries. The main technique applied to the computation is the combination of a linear grading function and the method of multiple Lie brackets. We introduce new notations for block matrices to simplify the expression of block matrices. The new notations help to prove the non-degeneracy of huge size matrices by an induction technique.