 Nonunique weights in a closed Bessel integral representation

Choose the square-root branch asymptotic to $\zeta$ outside the cut $[-i,i]$. The Laurent series of its reciprocal gives the displayed integral by termwise residue calculation. Adding any entire function to the weight leaves the integral unchanged, since its product with $e^{z\zeta }$ is entire. Thus a nonzero solution represented by a contour integral does not by itself determine the weight uniquely; an inference from the differential equation to an equation for the weight needs an injective representation or an explicit construction assumption.