  In this work we prove the uniqueness of solutions to the nonlocal linear
equation $L \varphi - c(x)\varphi = 0$ in $\mathbb{R}$, where $L$ is an
elliptic integro-differential operator, in the presence of a positive solution
or of an odd solution vanishing only at zero. As an application, we deduce the
nondegeneracy of layer solutions (bounded and monotone solutions) to the
semilinear problem $L u = f(u)$ in $\mathbb{R}$ when the nonlinearity is of
Allen-Cahn type. To our knowledge, this is the first work where such uniqueness
and nondegeneracy results are proven in the nonlocal framework when the
Caffarelli-Silvestre extension technique is not available. Our proofs are based
on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and
Valdinoci for nonlinear problems in dimension two.
