In the book called "Chaotic Worlds: from Order to Disorder in Gravitational N-Body Dynamical Systems", link here, the author states that if the dynamical system has an integral of motion which is not stationary along the periodic orbit, the monodromy matrix has a unit eigenvalue. Check here to see what a monodromy matrix is.
So, assuming we have a periodic orbit with initial conditions $x_0 = [r_0, v_0]^T$. Is the Jacobi constant which is the integral of motion of the CR3BP dynamical system constant?
My intuition says the the Jacobi constant is stationary as it is calculated from the initial conditions of the periodic orbit but why do we get unity eigenvalues.