In　this　paper,　the　unique　normal　form　for　a　class　of　three-dimensional　nilpotent　vector　fields　with　symmetry　is　investigated.　The　key　technique　used　is　a　combination　of　multiple　Lie　brackets,　linear　grading　function,　new　notations　of　block　matrices　and　first　integral　of　the　linear　part　of　the　given　vector　field　which　avoids　complicated　calculations.　The　first　and　the　second　order　normal　forms　are　computed　successively,　and　the　uniqueness　of　the　second　order　normal　form　is　proved　under　certain　conditions.