  We shall study the solvability of pseudodifferential operators which are not
of principal type. The operator will have real principal symbol and we shall
consider the limits of bicharacteristics at the set where the principal symbol
vanishes of at least second order. The convergence shall be as smooth curves,
then the limit bicharacteristic is a smooth curve. We shall also need uniform
bounds on the curvature of the characteristics at the bicharacteristics, but
only along the tangents of a given Lagrangean manifold. This gives uniform
bounds on the linearization of the normalized Hamilton flow on the tangent
space of this manifold at the bicharacteristics. If the quotient of the
imaginary part of the subprincipal symbol with the norm of the Hamilton vector
field switches sign from $-$ to $+$ on the bicharacteristics and becomes
unbounded as they converge to the limit, then the operator is not solvable at
the limit bicharacteristic.
