Integrable systems appeared in physics long ago at the onset of classical dynamics with examples being Kepler's and other famous problems. Unfortunately, the majority of nonlinear problems turned out to be nonintegrable. In accelerator terms, any 2D nonlinear nonintegrable mapping produces chaotic motion and a complex network of stable and unstable resonances. Nevertheless, in the proximity of an integrable system the full volume of such a chaotic network is small. Thus, the integrable nonlinear motion in accelerators has the potential to introduce a large betatron tune spread to suppress instabilities and to mitigate the effects of space charge and magnetic field errors. To create such an accelerator lattice one has to find magnetic and electrtic field combinations leading to a stable integrable motion. This paper presents families of lattices with one invariant where bounded motion can be easily created in large volumes of the phase space. In addition, it presents 3 families of integrable nonlinear accelerator lattices, realizable with longitudinal-coordinate-dependent magnetic or electric fields with the stable nonlinear motion, which can be solved in terms of separable variables.

The beam-beam interaction in the Tevatron collider sets limits on bunch intensity and luminosity. These limits are caused by a tune spread in each bunch which is mostly due to head-on collisions, but there is also a bunch-to-bunch tune spread due to parasitic collisions in multibunch operation. We propose to compensate these effects with the use of a countertraveling electron beam, and we present general considerations and physics limitations of this technique.

The wake forces produced by a beam bunch can be reduced by making the vacuum chamber cross section axially asymmetric. Furthermore, the asymmetry results in a betatron tune shift for particles in the tail of the bunch. As a result, transverse instabilities of the bunch should be significantly suppressed for an asymmetric vacuum chamber. [S0031-9007(99)08709-8] PACS numbers: 29.27.Bd, 29.20.Dh An ultrarelativistic charged particle generates electromagnetic fields behind it in the vacuum chamber. The net effect of these fields on a following charge is determined by integrating the force over a structure period of the vacuum chamber L. The integrated transverse force F caused by a slight offset r 0 of the leading particle from the axis of a round chamber is conventionally expressed in terms of the wake function [1]: Z L

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