  An alternative representation for the vector potential of the nonlinear
magnetic insert for the Integrable Optics Test Accelerator (IOTA), first
described in Sec. V.A. of the paper of Danilov and Nagaitsev, is determined
from first principles using standard complex variable methods. In particular,
it is shown that the coupled system consisting of the 2D Laplace equation and
the Bertrand-Darboux equation is equivalent to a single ordinary differential
equation in the complex plane, and a simple solution is constructed. The
results are consistent with the paper of Danilov and Nagaitsev, and this
concise representation provides computational advantages for particle tracking
through the nonlinear insert by avoiding numerical errors caused by small
denominators that appear when evaluating transverse derivatives of the vector
potential near the midplane. A similar representation is provided for the
spatial dependence of the two invariants of motion.
