According to Alexander's law (AL), the slow phase velocity of nystagmus of vestibular origin is dependent on horizontal position, with lower velocity when gaze is directed in the slow compared to the fast phase direction. Adaptive changes in the velocity-to-position neural integrator are thought to cause AL. While these changes have been described for the horizontal neural integrator, nystagmus often includes vertical and torsional components, but the adaptive abilities of the vertical and torsional integrators have not been investigated. We measured 11 patients with a peripheral vestibular asymmetry, and used second order equations to describe how velocity varied with position. Horizontal velocity changed with horizontal position in accordance with AL, and the second order term for horizontal position was also significant. While velocity decreased in the slow phase direction, it was relatively unchanged more than 10° into the fast phase direction. Vertical velocity was also highest in the vertical fast phase direction, and the second order term for vertical position was also significant, as vertical velocity increased in the vertical fast phase direction, but was unchanging in the slow phase direction. Torsional velocity varied linearly with horizontal, but not vertical, position. These results show that the horizontal and vertical oculomotor neural integrators react to altered vestibular input by maintaining different integrating time constants depending upon gaze direction.