One Variable Equations in Torsion-Free Hyperbolic Groups

Let $\mathrm{\Gamma }$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $\mathrm{\Gamma }$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $\mathrm{\Gamma }$ is free.