The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree K²=1

We undertake a study of topological properties of the real Mordell-Weil group MWR of real rational elliptic surfaces X which we accompany by a related study of real lines on X and on the "subordinate" del Pezzo surfaces Y of degree 1. We give an explicit description of isotopy types of real lines on YR and an explicit presentation of MWR in the mapping class group Mod(XR). Combining these results we establish an explicit formula for the action of MWR in H₁(XR).