Summary: The present paper is a continuation of M. Sh. Birman and T. A. Suslina [Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra Anal. 9, No. 1, 32-48 (1997); translated in St. Petersbg. Math. J. 9, No. 1, 21-32 (1998; Zbl 0890.35096)]. Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, Algebra Anal. 10, No. 4, 1-36 (1998); translated in St. Petersbg. Math. J. 10, No. 4, 579-601 (1999; Zbl 0922.35101] Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, Algebra Anal. 11, No. 2, 1-40 (1999); translated in St. Petersbg. Math. J. 11, No. 2, 203-232 (2000; Zbl 0941.35015)]. A two-dimensional periodic magnetic Schrödinger operator with variable metric is considered. The electric potential is assumed to contain a term proportional to the \(\delta\)-function supported on a periodic system of piecewise-smooth curves. It is shown that, under rather general assumptions on the problem data, the spectrum of the Schrödinger operator is absolutely continuous.