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.

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