The general-linear-complex singularity of Stewart mechanism is a very important problem in the parallel manipulator. Its general regularity is not found yet during the past two decades. StOnge and Gosselin pointed out that the singularity locus of the Stewart mechanism at some given orienb-tions of the moving platform should be polynomial expressions with varied degrees in 2000, but they didn’t formulate the expression. Based on the kinematics singularity principle and the geometry cond-tion proposed by Huang Zhen in 1999, firstly the singularity equation in degree two is derived. It is a hyperbola when the orientation of the moving platform is given. This result is also proved using screw theory. Then some singularity surfaces are gotten in three-dimensional space. This result is of impcr-tant significance. The general-linear-complex singularity of Stewart mechanism is a very important problem in the parallel manipulator. Its general regularity is not found yet during the past two decades. StOnge and Gosselin pointed out that the singularity locus of the Stewart mechanism at some given orienb -tions of the moving platform should be polynomial expressions with varied degrees in 2000, but they did not formulate the expression. Based on the kinematics singularity principle and the geometry cond-proposed by Huang Zhen in 1999, first the singularity equation in degree This result is also proven using screw theory. Then some singularity surfaces are gotten in three-dimensional space. This result is also proved by screw theory. Then result is of impcr-tant significance.