| In This paper we consider a class of operators with multiple characteristics:P(x,D)=D14 x14D24-(i1/2 (-i)1/2)D12D2 4x1D1D22-i(i1/2-(-i)1/2)x12D23 (1 2i)D22 C. Generally P(x,D) = LtL(x,D) C where L is a operator which has no solutions. We conclude that the lower order terms, including the zero order term, influence essentially the local solvability, the principal part p4(x,D)=D14 x14D24 of p(x,D) is a solvable operator in a neighbourhood of origin, p(x,D) isn't solvable when C = 0, and p(x,D) becomes again solvable when C>0. We also discuss Грущин operator with zero term. Finally, we prove that if the nonhomogeneous term f(x1,x2) is |x1|ψ＇(x2)(ψ is a real function), there are solutions if and only if ψ(x2) is analytic. |
