  We give an explicit description of adjoint quotient maps for Jacobson-Witt
algebra Wn and special algebra Sn. An analogue of Kostant's differential
criterion of regularity is given for Wn. Furthermore, we describe the fiber of
adjoint quotient map for Sn and construct the analogs of Kostant's transverse
slice. In addition, we analysis the nilpotent elements and generalize some
Premet's results to Sn.
