A Bargmann system and the involutive representation of solutions of the Levi hierarchy

The author considers a linear spectral problem associated with the Levi hierarchy of equations. A Bargmann system under the so-called Bargmann constraint between the eigenfunctions and potentials is proposed, and is proved to be a completely integrable Hamiltonian system in the Liouville sense. Moreover, by the involutive solution of the compatible systems under the Bargmann constraint he presents the involutive representation of solutions of the Levi hierarchy. Especially, he obtains the involutive representation of the solution of the well known Burgers equation u t =-u xx +2uu x .