  We develop the method of regularized moving frames of Fels and Olver to
obtain explicit general formulas for the basis invariants that generate all the
joint differential invariants, under gauge transformations, for the operators
[\o{L}=D_xD_y +a(x,y) D_x + b(x,y) D_y +c(x,y)] and an operator of arbitrary
order.
  The problem appeared in connection with invariant construction of Darboux
transformations for $\o{L}$.
