The mechanism for stable gaze-holding requires a neural integrator that converts pulse of neural discharge to steady firing rate. The integrator is feedback-dependent, impaired feedback manifests as either “unstable” integration when it is too much or “leaky” when it is too little. The “unstable” integrator is known to cause sinusoidal oscillations of the eyes called pendular nystagmus, whereas the “leaky” integrator causes jerky eye oscillations called gaze-evoked nystagmus. We hypothesized that integrator can be simultaneously leaky and unstable. Mechanistically, some parts of network are served by increased feedback gain (unstable network), while other part would be decreased feedback gain (leaky). Both leaky and unstable, the network converges on the ocular motor plant, leading to simultaneously present gaze-evoked jerk and sinusoidal nystagmus. We tested our hypothesis by measuring eye movements with search coil technique in 7 multiple sclerosis patients. Five of these patients had gaze-evoked nystagmus and superimposed pendular nystagmus. The gaze-evoked nystagmus depicted all the features of “leaky” integrator, that is, the drifts were always toward the null that was located at the central eye-in-orbit orientation, there were no drifts at null, and the drift velocity increased as the eyes moved farther away from the null. The pendular nystagmus had all the features of “unstable” integrator, that is, constant 4- to 6-Hz frequency, eye-in-orbit position dependence of the oscillation amplitude, and the voluntary saccade causing an oscillatory phase reset. These features were then simulated in a computational model conceptualizing our hypothesis of simultaneously leaky and unstable neural integrator.