  A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder
continuous functions defined on a contour $\Gamma$ that is the pullback of
$\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus
1, is introduced and studied, and its relations with holomorphic
$\Sigma$-factorization are discussed. It is applied to study and solve some
scalar Riemann-Hilbert problems in $\Sigma$ and vectorial Riemann-Hilbert
problems in $\mathbb{C}$, including Wiener-Hopf matrix factorization, as well
as to study some properties of a class of Toeplitz operators with $2 \times 2$
matrix symbols.
