  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/or electric 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 two examples 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.
