  In this article, we investigate the transverse dynamics of a single particle
in a model integrable accelerator lattice, based on a McMillan
axially-symmetric electron lens. Although the McMillan e-lens has been
considered as a device potentially capable of mitigating collective space
charge forces, some of its fundamental properties have not been described yet.
The main goal of our work is to close this gap and understand the limitations
and potentials of this device. It is worth mentioning that the McMillan axially
symmetric map provides the first-order approximations of dynamics for a general
linear lattice plus an arbitrary thin lens with motion separable in polar
coordinates. Therefore, advancements in its understanding should give us a
better picture of more generic and not necessarily integrable round beams. In
the first part of the article, we classify all possible regimes with stable
trajectories and find the canonical action-angle variables. This provides an
evaluation of the dynamical aperture, Poincar\'e rotation numbers as functions
of amplitudes, and thus determines the spread in nonlinear tunes. Also, we
provide a parameterization of invariant curves, allowing for the immediate
determination of the map image forward and backward in time. The second part
investigates the particle dynamics as a function of system parameters. We show
that there are three fundamentally different configurations of the accelerator
optics causing different regimes of nonlinear oscillations. Each regime is
considered in great detail, including the limiting cases of large and small
amplitudes. In addition, we analyze the dynamics in Cartesian coordinates and
provide a description of observable variables and corresponding spectra.
