  Nonlinear, completely integrable Hamiltonian systems that serve as blueprints
for novel particle accelerators at the intensity frontier are promising avenues
for research, as Fermilab's Integrable Optics Test Accelerator (IOTA) example
clearly illustrates. Here, we show that only very limited generalizations are
possible when no approximations in the underlying Hamiltonian or Maxwell
equations are allowed, as was the case for IOTA. Specifically, no such systems
exist with invariants quadratic in the momenta, precluding straightforward
generalization of the Courant-Snyder theory of linear integrable systems in
beam physics. We also conjecture that no such systems exist with invariants of
higher degree in the momenta. This leaves solenoidal magnetic fields, including
their nonlinear fringe fields, as the only completely integrable static
magnetic fields, albeit with invariants that are linear in the momenta. The
difficulties come from enforcing Maxwell equations; without constraints, we
show that there are many solutions. In particular, we discover a previously
unknown large family of integrable Hamiltonians.
