= Solution
The jump coefficient $G(t)=(t^2-1)/t$ has one enclosed pole at zero and no enclosed zeros, because $\pm1$ are outside. Its winding index is therefore \b[$\kappa=-1$]. A homogeneous canonical pair is
$$
\boxed{X_+(z)=z^2-1,\qquad X_-(z)=z,\qquad X_+/X_-=G.}
$$
Each is analytic and nonvanishing in its required finite domain; the exterior growth $X_-\sim z$ encodes the negative index. Dividing the unknown by these factors gives an additive jump
$$
D_+-D_-=g(t),\qquad g(t)=\frac{t-1+K}{2t(t^2-1)}.
$$
The Cauchy-integral solution for $D$ normally has an exterior $z^{-1}$ coefficient $-(2\pi i)^{-1}\int_Lg$. Since $C_-=zD_-$ must decay, this coefficient must vanish. Thus the solvability condition is $\int_Lg(t)dt=0$. Its only enclosed pole is zero, with residue $(1-K)/2$, so
$$
\boxed{K=1.}
$$
These are the <negative-index scalar Riemann-Hilbert moment conditions> for index minus one. There is no arbitrary <polynomial> addition: an entire addition respecting $D_-=O(z^{-2})$ is zero by Liouville's theorem.
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