  An integral equation reformulation of the Maxwell transmission problem is
presented. The reformulation uses techniques such as tuning of free parameters
and augmentation of close-to-rank-deficient operators. It is designed for the
eddy current regime and works both for surfaces of genus $0$ and $1$.
Well-conditioned systems and field representations are obtained despite the
Maxwell transmission problem being ill-conditioned for genus $1$ surfaces due
to the presence of Neumann eigenfields. Furthermore, it is shown that these
eigenfields, for ordinary conductors in the eddy current regime, are different
from the classical Neumann eigenfields for superconductors. Numerical examples,
based on the reformulation, give an unprecedented $13$-digit accuracy both for
transmitted and scattered fields.
