In this paper, we consider quasi-periodically forced, restricted \((1+1)\) -vortices systems with Hamiltonians of the form \(\begin{aligned} \Psi (t,x,y)=\frac{1}{2}\ln (x^2+y^2)+P(\omega t,x,y),~ \end{aligned}\) where one of point vortices is fixed at origin (0, 0), P is real analytic in \(\mathbb {T}^d\times \mathbb {R}^2\) , and \(\omega \) is either Diophantine- or Brjuno-like. Using the KAM method, we show the existence of a family of invariant cylinders that surround the origin and consist of quasi-periodic motions, implying the stability of the singularity (0, 0) in the sense of Lyapunov. In the case that P is perturbative, we also give various conditions for the existence of response quasi-periodic solutions with large amplitudes. Several applications to point-vortex problems arising in fluids and super-conductivity are also discussed.
