I reviewed these basics as described by Marion and Thornton. The number of proper Lagrange variables is equal to the number of degrees of freedom, but it is also possible to use the method of undetermined multipliers when the number of Lagrange variables is greater than the number of degrees of freedom. However, a simple solution to the problem is to solve Eqs. (17) to (19) of Note 371(2) simultaneously, to give r1, r2 and r3 in terms of theta, phi and chi. These are the required orbits. There are three dimensions (degree of freedom) and three proper Lagrange variables, r1, r2, and r3. So there are three differential equations in three unknowns, an exactly determined problem. The orbits are r1(theta, phi, chi), r2(theta, phi, chi) and r3(theta, phi, chi). Finally use

r squared = r1 squared + r2 squared + r3 squared

to find r(theta, phi, chi) and its precessions. In the planar limit it should reduce to a conic section without precession.