In this paper, I mainly prove the following results. For every energy value below the minimum of the first, second and third critical value, each bounded component of the regularized energy hypersurface of the Lagrange problem with \(0<m_2\le m_1\le 9m_2\) , \(m_1\ge \frac{\epsilon }{2}>0\) and \(m_2\ge \frac{3\epsilon }{8}\) arises as the boundary of a strictly monotone toric domain, which is dynamically convex as a corollary. For the Euler problem as a special case of the Lagrange problem, when the energy \(c<-m_1-m_2\) , the bounded component around the fixed center e of the regularized energy hypersurface of the Euler problem with two fixed points e and m of masses \(m_1\) and \(m_2\) respectively satisfying \(m_1>0, m_2\le 0\) and \(m_1\ge |m_2|\) arises as the boundary of a convex toric domain. Together with Gabriella Pinzari’s result, when the energy is less than the critical value, the toric domain \(X_{\Omega _{m_2}}\) defined above is concave for \(m_2\ge 0\) , convex for \(m_2\le 0\) .
