Monodromy of a Class of Logarithmic Connections on an Elliptic Curve

The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-$2$ double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their monodromy, differential Galois group and the underlying rank-$2$ vector bundle. The latter is described in terms of elementary transforms. The question of its (semi)-stability is addressed.