On the properties of the identically singular Lagrange problem

We consider the identically singular Lagrange problem of the calculus of variations. It is investigated how the conjugate points and the Jacobi equation index are related to the solvability conditions of the appropriate matrix Riccati equation and the reducibility of the functional to a perfect square. The criterion of the trajectory variation smallness at a small variation of the functional in the neighborhood of the extremal is obtained.