With the development of soliton theory, Lie algebra shows great promise. As it is well known, professor Tu presented a method which used Lie algebra as a tool to construct systematically the Lax equation. Especially, Tu^[1] obtained trace identity-based loop algebra A₁. Later, many kinds of Lie algebras and their subalgebras were constructed, and some hierarchies with physics meanings are given.^{[2– 11]} Recently, professor Ma^[12] generated a hierarchy of soliton equations associated with the real Lie algebra so(3, ℝ ) and shows that each equation has a bi-Hamiltonian structure and integrable in the Liouville sense. The real Lie algebra of the so(3, ℝ ) is not isomorphic to the real Lie algebra sl(2, ℝ ) over the real field ℝ , and the soliton hierarchy^[12] is not gauge-equivalent to the famous AKNS soliton hierarchy over the real field ℝ . In fact, the two Lie algebras are isomorphic to each other over the complex field , thus hierarchy^[12] is equivalent to the AKNS hierarchy over the complex field .

It is known that Darboux transformation provides us with a purely algebraic and powerful method to find a new solution of a nonlinear equation from a known solution in Refs.  [13]– [16]. The classical Darboux transformation^[17] mainly deals with a Sturm– Liouville problem, which presented solutions to the Korteweg-de Vries hierarchy of (1 + 1)-dimensional integrable systems with new potentials and new wave functions. Based on the work, proper transformation that applies to (2 + 1)-dimensional integrable systems was found. The method of the Darboux transformation has also been applied to the discrete integrable systems successfully. The present paper aims to review application of the Darboux transformation technique for superintegrable systems. In general, we assume every element of T is polynomial with respect to λ .

In this paper, we first introduce a Lie algebra of the special orthogonal group, g = so (4, ), which is a 4× 4 trace-free, skew-symmetric complex matrix set. As its application, we obtain a soliton hierarchy which is reduced to AKNS hierarchy and presented its Hamiltonian structures and Liouville integrability. Furthermore, for one of the equations in the resulting hierarchy, we present the Darboux matrix T, which the element is related to the spectral parameter λ according to the different position and number of spectral parameter λ .

We consider the spectral problem  (4) by introducing the temporal part

where

The compatibility condition ϕ _xt = ϕ _tx yields a zero curvature equation

Now we consider a Darboux transformation

where T is defined by

and

A new spectral problem reads

where U and N has the same form as U and N, except replacing p, q, r, and s with p, q, r, and s.

Now we consider the basic forms of T. First of all, we assume

where a_ij and b_ij (i, j = 1, 2, 3, 4) are functions of x and t. By inserting Eq.  (27) into Eq.  (24), we read that the coefficients matrix of λ ² are

In the following, we consider the simplest non-trivial forms of T

As the subalgebras of so(4, ), G₁ = span{e₁, e₂, e₃} and G₂ = span{e₁, e₅, e₆}, both are isomorphic to so(3, ), and both are isomorphic to sl(2, ) over the complex field , we know that equations  (16) and (17), as the reductions of Eq.  (11), are equivalent to AKNS hierarchy over the complex field . Therefore, equation  (9) was regarded as the integrable coupling of AKNS and itself, which was also called the bi-AKNS system.

In the past, general coupling form_{[18– 23]} by making use of Lie algebra A₁, A₂ and their subalgebras was presented in the following

and general super-integrable hierarchy form through use of Lie subalgebra B(0, 1) was proposed by Li, ^[24] Ma^[25] as follows:

In this paper, we obtained the integrable coupling by making use of orthogonal Lie algebra so(4, ), which possessed the form of Eq.  (96), and is not super-integrable coupling. It is the first time that we have found the results.