Algebraic cycles satisfying the Maurer-Cartan equation and the unipotent fundamental group of curves

We address the question of lifting the etale unipotent fundamental group of curves to the level of algebraic cycles and show that a sequence of algebraic cycles whose sum satisfies the Maurer-Cartan equation would do the job. For any elliptic curve with the origin removed and the curve , we construct such a sequence of algebraic cycles whose image under the cycle map gives rise to the etale unipotent fundamental group of the curve.