Integrable cases in nonlinear Betatron motion

The integrability of the one dimensional betatron equation of motion is discussed. Although it is known that the general time dependent differential equation of second order describing single particle motion in presence of sextupoles is nonintegrable, special cases may be found in which integrability can be proven and first integrals can be written analytically. The present paper introduces a method to find specific sextupole distributions for which integrability can be obtained. The solutions of such integrable equations are investigated numerically and analytically and a rigorous stability analysis of the solutions with respect to their initial conditions is performed. The possibilities of applications of this theory to real machines is discussed