  We are interested in studying moduli spaces of rank 2 logarithmic connections
on elliptic curves having two poles. To do so, we investigate certain
logarithmic rank 2 connections defined on the Riemann sphere and a
transformation rule to lift such connections to an elliptic curve. The
transformation is as follows: given an elliptic curve $C$ with elliptic
quotient $\pi\colon C\to{\mathbb P}^1$, and the logarithmic connection
$(E,\nabla)$ on ${\mathbb P}^1$, we may pullback the connection to the elliptic
curve to obtain a new connection $(\pi^*E, \pi^*\nabla)$ on $C$. After suitable
birational modifications we bring the connection to a particular normal form.
The whole transformation is equivariant with respect to bundle automorphisms
and therefore defines a map between the corresponding moduli spaces of
connections. The aim of this paper is to describe the moduli spaces involved
and compute explicit expressions for the above map in the case where the target
space is the moduli space of rank 2 logarithmic connections on an elliptic
curve $C$ with two simple poles and trivial determinant.
