In classical integrable systems, the term complete integrability is well defined. More precisely in n degrees of freedom, a Hamiltonian system can be integrated up to quadrature if there exist n involutive functions, which are functionally independent. When taking one of these functions as the Hamiltonian H and thinking of others as its first integrals, this Hamiltonian is said to be completely integrable in the Liouville sense. Even though the number of independent functions, which are in involution is n at most, maybe additional integrals of the Hamiltonian H exist and they will definitely generate a non-Abelian algebra of integrals of H. In general, these integrals of motion do not yield finite-dimensional Lie algebras, but more complicated algebraic structures, namely, Possion algebra. These systems, with more integrals, are called super-integrable. The maximal number of further independent integrals is . If there are exactly independent first integrals, we say that the Hamiltonian H is maximally superintegrable. The isotropic harmonic oscillator, the Kepler system, and the Calogero-Moser system are the well-known examples. Maximally superintegrable systems have some interesting features. For instance, they can be separable in more than one coordinate system and all bounded classical trajectories are closed.

There are numerous papers on superintegrable systems, both classical and quantum, see Refs. [1–5] and references therein. In most of these work, it was restricted to quadratic integrals of motion. The quadratic integrals have an intimate connection with the separation of variables in the Hamilton-Jacobi and Schrödinger equations and generally they produce so-called quadratic algebras.^[6]

In contrast, much less attention has been devoted to integrable and superintegrable systems with cubic and higher-order integrals of motion. Ten different potentials in a complex Euclidian plane, which admitted a cubic integral of motion are listed in Ref. [7]. Among them, seven are reducible for their cubic integrals can be written as the Poisson commutators of two quadratic integrals.^[8] Systematic studies for superintegrable systems with at least one cubic integral was initiated in Ref. [9]. There the coexistence of first- and third-order integrals of motion in two-dimensional classical and quantum mechanics are considered. Corresponding potentials and integrals are identified. The above mentioned paper was followed by a series of publications. In Refs. [10–11], superintegrable systems that are separable in Cartesian coordinates and admit a cubic integral in classical and quantum mechanics are presented. Polynomial Poisson algebras for classical superintegrable systems with a cubic integral are given in Ref. [12]. The complete classification of quantum and classical superintegrable systems allowing the separation of variables in polar coordinates and admitting an additional integral of motion of order three in the momentum is performed.^[13] In Ref. [14], the conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied.

In Ref. [15], families of Hamiltonians of the form

Here we deal with the Hamiltonian functions taking the form of

Now we recall some known facts on Killing vectors. In a Riemannian manifold , a Killing vector field X is the infinitesimal generator of a symmetry of the metric g. Namely X is a generator of isometries. In geometric terms X satisfies the condition with the Lie derivative . For a manifold of dimension n, the metric possesses linearly independent Killing vectors at most. Note that when the space is either flat or constant curvature, it admits the maximal group of isometries, which is dimension.

For a Riemannian manifold , which corresponds to a 2n-dimensional configuration space, with coordinates (), of a system, g determines a kinetic Lagrangian such that the associated motion is just the geodesic motion with kinetic energy , where g^ij is the inverse of the metric tensor g_ij when g_ij is nonsingular. Note that is generalised position (this can be for example Cartesian coordinates, angles, arc lengths along a curve) and is corresponding generalised momentum. For a metric with isometries, the Killing vectors correspond to functions that are linear in momenta (the so-called Noether integrals) and that Poisson commute with the kinetic energy H₀.

Following the standard approach, direct computation shows that the kinetic energy

In view of the algebraic structure, the quadratic Casimir of the Euclidean algebra (3) is . And the corresponding Killing vectors can be written as

Generally speaking, for the kinetic energy, the introduction of a potential will destroy its first integrals. Nevertheless, it is well-known that for a flat metric, the leading-order terms of all second- and higher-order integrals of motion are determined by corresponding Killing tensors built from tensor products of Killing vectors.[18]

Applying the canonical transformation governed by the generating function

And thus, for the Hamiltonian (1), we should consider the cubic integrals

With integrable systems (4), (5), and (6) allowing one cubic first integrals in hand, we impose a second independent function, which is the second order of motion and Poisson commutes with its Hamiltonians. Here we consider superintegrability of systems (4), (5), and (6) separately.

4.1 Superintegrability of System (4)

(i) Superintegrable Restriction with

Given the second first integrals, the arbitrary function ψ in Eq. (4) is specified up to finite parameters. In the case of

the relation results in

With these potentials, we have and . is a quartic integral and it may be a polynomial combination of H, F₁, and F₂. However, it is not the case. We try to close the algebra at lowest dimension possible. Now we add two elements K₂ and F₃, with

According to the nonzero commutation relations

the integrals H, K₂, F₁, F₂, and F₃ form a polynomial Poisson algebra. Following the Poisson relations and without using the specific representation, the Casimir functions and can be obtained. Inserting the exact forms of the integrals, we find that

which are the algebraic relations among the five first integrals H, K₂, F₁, F₂, and F₃.

According to , the cubic integral F₁ can be expressed as the commutator of two quadratic integrals F₂ and F₃. Consequently, the cubic integral is reducible. And the superintegrable system coincide with system (14) obtained in Ref. [16].

(ii) Superintegrable Restriction with

Assuming system (4) allows the following quadratic integral

leads to

It can be easily seen that . Then we find

which shows that the functions H, F₁, F₂, K₂ generate a Poisson algebra. And of course, they are dependent and satisfy the relation .

(iii) Superintegrable Restriction with

Given a quadratic integral of the form

and presume , we arrive at

It can be verified that the integrals H, K₂, F₁, F₂ give a Possion algebra because of

And for this algebra, the constraint holds.

4.3 Superintegrability of System (6)

System (6) is superintegrable if the Hamiltonian there admits another independent first integral. Choosing , we have

Moreover . And thus H, F₁, F₂ generate a Poisson algebra.

5 Integrability of Eq. (1) When

Provided n = 2, the kinetic energy (2) reduces to . Its Killing vectors take the forms of

Imposing the cubic first integral on

We now look for superintegrable systems (8) with the potential given in Eq. (9) and prove that systems (8) corresponding to Eqs. (10) and (11) are superintegrable themselves.

To set and suppose it is admitted by Eq. (9) give

For system (10), it also possesses a quadratic first integral

Analogously, the quadratic integral is admitted by the Hamiltonian (8) with (11) and the corresponding Poisson algebra is governed by H, F₁, F₂, which satisfy .

In this paper we dealt with the super integrability of some Hamiltonian systems, namely for 2n-dimensional Hamiltonian equation, there exists more than n first integrals. Such a property is stronger compared with “complete integrability”, namely the solution structure of such systems has fewer arbitrary parameters since the motion is more restricted by the extra first integrals. Here we constructed some superintegrable systems with one quadratic and one cubic integrals of motion, and build up Poisson algebra for each superintegrable Hamiltonian system. Moreover, the algebraic dependence relations to each Poisson algebra are also given.

A generalization of integrable Hamiltonian equations is the study of the integrability of infinite-dimensional equations, namely integrable partial differential equations (also known as soliton equations) including a number of (1+1)- and (2+1)-dimensional equations^†, in which case an infinite number of independent first integrals being in involution is required in order to guarantee the integrability, see e.g. Adler,^[20] Gel’fand and Dikii,^[21] Magri,^[22] Fuchsteiner and Fokas,^[23] and TAH,^[24] etc.

The theory of the completely integrability of soliton equations is also applicable to the so-called super soliton equations. We refer the reader to Zhang^[25–27] and references therein.