Uniqueness in a Characteristic Cauchy Problem with Variable Multiplicity

For the operator P(c1, c2) = (∂x + a1xk∂y + a2xk)(∂x − b1xk∂y − b2xk) − c1xk−1∂y − c2xk−1 with a1•b1>0, a2, b2 ∈ R, c1, c2 ∈ C, we show that the local solutions of the (characteristic) Cauchy problem with data u, uy at y = 0, are unique if and only if the pair (c1, c2) does not belong to a discrete set, namely if (c1, c2) ≠ [j(k+1)+l]•(a1+b1, a2+b2), for all j ∈ Z, l = 0, 1.