  Proton accumulator rings and other circular hadron accelerators are
susceptible to intensity-driven parametric instabilities because the
zero-current charged particle dynamics are characterized by a single tune.
Landau damping can suppress these instabilities, which requires energy spread
in the beam or introducing nonlinear magnets such as octupoles. However, this
approach reduces dynamic aperture. Nonlinear integrable optics can suppress
parametric instabilities independent of energy spread in the distribution,
while preserving the dynamic aperture. This novel approach promises to reduce
particle losses and enable order-of-magnitude increases in beam intensity. In
this paper we present results, obtained using the Lie operator formalism, on
how chromaticity and dispersion affect particle orbits in integrable optics. We
conclude that chromaticity in general breaks the integrability, unless the
vertical and horizontal chromaticities are equal. Because of this, the
chromaticity correcting magnets can be weaker and fewer correcting magnet
families are required, thus minimizing the impact on dynamic aperture.
