Elliptic bob path at zero horizontal momentum ($(X-X_c{)}^2/[l{m}_1/(m_1+m_2){]}^2+Y^2/l^2=1$)

Zero horizontal momentum fixes the horizontal centre of mass. The bob coordinates become $X=X_c+l{m}_1\sin \varphi /(m_1+m_2)$ and $Y=-l\cos \varphi$. Eliminating the angle gives an ellipse; an ordinary oscillation traverses only its permitted arc. The support counter-moves, so the bob's path differs from the circle about a fixed pivot.