Asymptotics of the solution of the system $A(x,-ih\frac\partial{\partial x})$ as $h\to0$ in the case of characteristics of variable multiplicity

In this paper we develop a new asymptotic method for pseudodifferential operators in the case of characteristics of variable multiplicity; the $N$th term of the asymptotics is expressed in terms of an $N$-dimensional integral of a rapidly oscillating function of $(N+n)$ arguments, where $n$ is the dimension of the space ($x=x_1,\dots,x_n$).