3.3. The 2-sphere

The unit two-dimensional sphere S<sup>2</sup> ⊂R<sup>3</sup> admits a completely integrable geodesic flow. Indeed, the geodesic flow of an ellipsoid is also completely integrable with the second integral of motion that is, in general, a quadratic form in the momenta.

The fundamental problem is to describe the moduli space of completely integrable Hamiltonians on T� S2 . The sub-problem of describing the integrable Riemannian (resp. natural or mechanical) Hamiltonians H has received wide-spread attention. When H is Riemannian, the most common approach is to assume the second integral F is polynomial-in-momenta, and without loss of generality, homogeneous. If the degree of F is fixed, then the problem of determining H&F is reducible to a non-linear PDE in the coefficients of F. When the degree is 1, the first integral F is a momentum map of a T<sup>1</sup> isometry group (see below). When the degree is 2, then the Hamiltonian is Liouville, a classical result due to Darboux c.f. [27]. In degree 3, there is the well-known case due to Goryachev-Chaplygin, and more recent cases due to Selivanova, Dullin and Matveev and Dullin, Matveev and Topalov and Valent [28–33]. In degree 4, Selivanova and Hadeler & Selivanova have produced a family of examples using the results of Kolokol'tsov [34, 27]. Beyond degree 4, Kiyohara has provided a construction of a smooth Riemannian metric H with an independent first integral F of degree k for any k≥1. In this construction, the metric H depends on a functional modulus, and so for each k, the set is infinite dimensional [35].

## 3.4. Super-integrable systems with a linear-in-momenta first integral

Let us review the work of Matveev and Shevchishin in more detail [36]. These authors impose an additional formal constraint that the metric possess one first integral that is linear-inmomenta. In conformal coordinates ðx, yÞ where H ¼ <sup>1</sup> 2cðxÞðp<sup>2</sup> <sup>x</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> <sup>y</sup>Þ, the existence of a cubic integral is reduced to a second-order ODE involving c.

From a geometric perspective, it is more natural to introduce coordinates adapted to the isometry group. That is, the existence of a linear-in-momenta first integral is equivalent to the existence of an isometry group containing T<sup>1</sup> . The action of T<sup>1</sup> on S<sup>1</sup> induces a cohomogeneity-1 structure. The fixed set of the <sup>T</sup><sup>1</sup> action is a set of points {p−, <sup>p</sup>þ} which are equidistant along any minimal geodesic; and the principal T<sup>1</sup> -orbits are orthogonal to these geodesics. If <sup>γ</sup> : <sup>½</sup>−T, <sup>T</sup>� ! <sup>S</sup><sup>2</sup> is a minimal geodesic such that <sup>γ</sup>ð�TÞ ¼ <sup>p</sup>�, then we can let <sup>ð</sup>r, <sup>θ</sup><sup>Þ</sup> be 'polar' coordinates adapted to this structure. The Hamiltonian H and polynomial-in-momenta integral F can be written in the adapted coordinates as

$$H = \frac{1}{2} \left( p\_r^2 + s(r) p\_\theta^2 \right), \\ F = e^{i v \theta} \times \sum\_{j=0} a\_j(r) p\_r^i p\_\theta^{N-j}, \tag{41}$$

where v∈Z, 3≤N is a positive integer and the coefficients aj are to be determined. The equation {H, F}≡0 is equivalent to a differential system that couples the coefficients a0, …, aN, s and an anti-derivative S of vs:

$$d\mathbf{S}\_{\parallel} = \begin{array}{c} \upsilon \mathbf{s} \,\mathrm{d}\mathbf{r}, \\\\ \end{array} \tag{42a}$$

$$\mathbf{d}a\_{j} = \begin{array}{c} 1 \\ 2 \end{array} \big( N + 2 \neg j \big) a\_{j-2} \operatorname{d}s \text{-} a\_{j-1} \operatorname{d}S; (j = 0, \ldots, N), \tag{42b}$$

$$\mathbf{ds} \quad = \begin{array}{c} \mathbf{2}v\mathbf{a}\_{\mathrm{N}}/a\_{\mathrm{N}-1}\,\mathrm{d}r \end{array} \tag{42c}$$

where a<sup>−</sup><sup>2</sup> ¼ a<sup>−</sup><sup>1</sup> ¼ 0. It is clear that the general solution of (42b), without the compatibility condition (42c), is obtained via repeated quadratures of products of s and S. The compatibility condition distinguishes those solutions which may arise from (41). The behaviour of s at <sup>r</sup> ¼ �<sup>T</sup> ultimately determines whether the solution obtained arises from a <sup>T</sup><sup>1</sup> -invariant Riemannian Hamiltonian H and an independent first integral F on T� S2 .

In case N ¼ 3, the differential system reduces to a third-order nonlinear ODE similar to that studied by Chazy, in his generalization of the Painlêvé classification ([37], Eq. (6)). Based on the work of Matveev and Shevchishin [36], we know the solutions to this equation are real-analytic and define a parameterized family of super-integrable Riemannian metrics with cubic-inmomenta first integral. The latter authors do not solve the ODE explicitly.
