= Solution
Set $\mu=GM>0$ and interpret $L$ in the displayed equation as <specific angular momentum>, $L=r^2\dot\psi$, rather than total <angular momentum>. The <gravitational potential> is per unit <mass> and $a$ has dimensions of length. <Spherical symmetry> fixes the orbital plane and conserves the vector <angular momentum>, while the autonomous <specific orbital energy> is also conserved.
For the <orbit equation for combined inverse-square and inverse-cube attraction>, define
$$
\nu^2=1-\frac{2\mu a}{L^2}.
$$
A regular nonradial bounded oscillation requires $L\ne0$ and $L^2-2\mu a>0$. Solving the linear <Binet equation> then gives
$$
\boxed{u(\psi)=\frac{\mu}{L^2-2\mu a}\left[1+e\cos\bigl(\nu(\psi-\psi_0)\bigr)\right].}
$$
For a noncircular bound <orbit> $0<e<1$; its <specific orbital energy> is $E=\mu^2(e^2-1)/[2(L^2-2\mu a)]$. The <effective potential> $[L^2-2\mu a]/(2r^2)-\mu/r$ has a <centrifugal barrier> and a minimum in this case. If $L^2-2\mu a\leq0$, the oscillator frequency is zero or imaginary and there is no regular finite-radius bound oscillation: the attractive centre lacks a <centrifugal barrier>. Unbound branches with $e\geq1$ are not closed ellipselike trajectories.
Successive pericentres differ in angle by $2\pi/\nu$. The full <phase space> state returns after $p$ radial oscillations and $q$ revolutions precisely when
$$
\boxed{\nu=\frac pq\in\mathbb Q_{>0},\qquad L^2=\frac{2\mu a}{1-p^2/q^2},}
$$
where $p,q$ are coprime positive integers; the second formula applies when $a\ne0$. Thus $a>0$ requires $0<p/q<1$ and $L^2>2\mu a$, whereas $a<0$ requires $p/q>1$. An irrational $\nu$ advances the apsidal direction through a dense set of angles and gives a nonclosed rosette, with only the four regular spherical invariants needed to label its two-dimensional closure.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-72-resonant-orbits.png]
{title=Closed rational-frequency orbits and a nonclosed irrational-frequency rosette in an inverse-cube-perturbed central force}
To see the additional resonant label explicitly, let $u_c=\mu/(L^2-2\mu a)$ and $w=u-u_c-i u'/\nu$. Then $w=C e^{i\nu(\psi-\psi_0)}$, so on a family with fixed resonant $L$,
$$
\boxed{w^q e^{-ip\psi}=C^q e^{-ip\psi_0}.}
$$
Its argument fixes the remaining apsidal phase modulo the symmetry of the closed curve; its magnitude is already determined by <specific orbital energy> and $L$. Choosing reference axes in the fixed orbital plane makes this single-valued under $\psi\mapsto\psi+2\pi$. This is the fifth independent <orbit> label on the resonant family described by <closed orbits with an inverse-cube perturbation>.
For $a\ne0$, $\nu$ varies with $L$. The <orbital resonance> condition holds only on selected <angular momentum> surfaces, so this construction is not a fifth global smooth integral on an open set of arbitrary nearby $L$. If the question asks for a maximally <superintegrable Hamiltonian system> on an open region of noncircular bound orbits, the required case is \b[$a=0$]: then $\nu=1$ for every $L$, and the conserved <Laplace-Runge-Lenz vector> $\mathbf A=\mathbf v\times\mathbf L-\mu\widehat{\mathbf r}$ fixes the apsidal direction globally. The relations $\mathbf A\cdot\mathbf L=0$ and $A^2=\mu^2+2EL^2$ leave five independent constants. For the usual orbit-wise interpretation, noncircular bound <orbits> with rational $\nu$ have five isolating labels; irrational $\nu$ has four. <Circular orbits> are closed for any allowed $\nu$ but have no independent apsidal phase, so the generic noncircular count cannot simply be applied to them.
For completeness, regular unbound branches are different. When $e\geq1$ and $\nu>0$, positive radius restricts the radial phase to $|\nu(\psi-\psi_0)|<\arccos(-1/e)\leq\pi$. The <apsidal invariant on an unbound central-force branch> is
$$
\exp\!\left(i\psi-\frac{i}{\nu}\operatorname{Arg}w\right)=e^{i\psi_0}.
$$
The principal argument is continuous on this scattering branch because it never reaches an apocentre across the negative-real-axis cut. Thus the periapsis direction can be a fifth single-valued scattering label for any positive $\nu$, including irrational $\nu$; the rationality condition applies to the noncircular bound case. For plunging or collision branches with $L^2-2\mu a\leq0$, local trajectory labels can still be defined away from the singular centre, but the preceding periodic-orbit criterion is inapplicable and no smooth continuation through $r=0$ is asserted.
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