is one of the difficult soliton equations, which has been paid common attention in physical and mathematical fields  - . In 1983, H. Flaschka put forward a problem of the third order differential operator associated with the Boussinesq Neumann system . Some works focus on the decomposition and the structures of the Modified Boussinesq equation    . The decomposition of the Boussinesq Neumann system has not been done thoroughly for a long time. A Neumann system of the Boussinesq equation associated with the third order differential operator is obtained in this paper, which is the extension of the famous KdV Neumann system associated with the second order differential operator. There are many methods to deal with the integrability and involutivity   . A generating function method starting from the Lax-Moser matrix  -  is used to give an effective way to prove the involutivity of integrals.
2. The Generating Function of Integrals
Let be the phase space. The canonical coordinates are
for short. Take , where are distinct spectral constants. We denote:
where X and Y are two N dimensional vectors.
For any matrix , the element of its adjoint is , where is the cofactor of the element in M.
In order to proof the Liouville integrability of the Hamiltonian system, consider the Lax-Moser matrix defined as:
Let . Then we have the generating function of integrals:
A series of polynomials , are defined as the coefficients of the power series expansions as :
The first few are expressed as :
By comparing the coefficients of the same power of , general explicit formulas are obtained:
Expand in non-negative power of :
as . By comparing the coefficients of the same power of , we have:
3. The Involutivity of Integrals
The involutivity is critical to the integrability of the Hamiltonian system, which is defined as the Poisson bracket of two conserved integrals being zero. A direct calculation gives:
Proposition 1. The Hamiltonian system for the -flow is expressed as:
Proof. From the Equation (3) and the property of Hamiltonian system, we calculate the partial derivatives of and with respect to , then the results are obtained.
Lemma 2. satisfies the Lax equation along the -flow:
Proposition 3. satisfies the Lax equation along the -flow:
Proof. can be expressed by as:
Resorting to Lemma 2, we have:
Lemma 4. The two determinants are true as :
From (6) and (7), we have:
where , and are polynomials of degree . From (8)-(10), another group of conserved integrals is obtained:
From the definition of involutivity of two conserved integrals and direct calcultion, we have:
Proposition 5. The integrals
are involutive in pairs:
4. Hamiltonian Systems
By direct calculations, the canonical equations of the Hamiltonian systems and can be expressed as:
where are matrices:
Proposition 6. Let be compatible solution of the and flow on the level set . Then the Boussinesq Equation (1) has a finite-parameter solution given as:
Proof. A direct calculation gives:
So from the above calculations and (16), we have:
which is exactly the “good” Boussinesq Equation (1).
The third order differential operator associated with the Boussinesq Neumann system is a critical point in researching problems of integrable system. In this paper, we obtain a Neumann system of the Boussinesq equation associated with the third order differential operator, which is the extension of the famous KdV Neumann system associated with the second order differential operator. By means of generating function from the Lax-Moser matrix, we prove the involutivity of integrals successfully. Meanwhile, a finite-parameter solution to the Boussinesq equation is obtained naturally.
This work is supported by the Foundation (Grant No. 11601123 and 202010463050).
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