  In this paper we present a method of constructing a nonlinear accelerator
lattice that has an approximate integral of motion that is given upfront. The
integral under consideration is a Hamiltonian in normalized (canonical)
coordinates that is preserved by a lattice with a given accuracy. We establish
a connection between the integrator of a Hamiltonian in normalized coordinates
and a real lens arrangement. We apply known algorithms of high-order symplectic
integrators, to produce several nonlinear lattices and show that this approach
could improve the design of the nonlinear insert considered at the IOTA and
UMER facilities. We also suggest new lattice design based on the Yoshida
integrator.
