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.　(C)　2014　Elsevier　Inc.　All　rights　reserved.