Integrability of Kepler Billiards at Zero-Energy

We consider a Kepler billiard with zero-energy in the plane defined inside a smooth closed connected simple curve which intersects all focused parabola at at most two points. {We show that} if has an invariant curve consisting of $2$-periodic orbits and there exists a $C^1$-first integral with non-vanishing gradient in the region between the invariant curve and the boundary curve, then the system is defined actually inside an ellipse with the Kepler center occupying one of the foci. This statement is obtained as a simple ``translation’’ of the theorem of Bialy-Mironov with Levi-Civita transformation.