I think that I.Krasil'schchik now the answer and many other information about these equations

These two papers may be useful to your problem:

1) N. Li, Q.P. Liu, Physics Letters A, 377, 257, 2012.

2) http://arxiv.org/abs/1309.3188v1

There is a rich theory devoted to the question of the integrability of dynamical systems. It was developed by Ziglin and the most powerful version was presented by Morales-Ruiz:

http://www2.caminos.upm.es/Departamentos/matematicas/morales%20ruiz/obsfin.pdf

There is as far as I know no better way to decide if a given dynamical system is integrable. If your equations of motion are polynomial and you have fixed a polynomial poisson structure, finding the corresponding Hamiltonian can be done by solving a system of linear equations by writing down the general Polynomial of the right degree and if you write down the Hamilton equations of motion and compare them with your PDE you get a system of linear equations.

If the Helmholtz conditions are fulfilled for your PDE system, then you can find a Hamiltonian vector field for this PDE system and also a Hamiltonian function. So, the PDEs can be written like Hamilton-Jacobi eqs. I thaink that, a very good book for this purpose is J. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry", Second Edition, Springer-Verlag, New York, Inc., 1999, or the book of P.J. Olver, Differential Geometry and Application to Lie Groups ...

The literature is vast. Have a look, for example, at arXiv:0812.4681 (for Camassa-Holm) and arXiv:1002.0077 (a detailed description of our approach with a lot of references).

A general approach consists of two steps: 1) find the lagrangian for the PDES; 2) construct the Hamiltonian through Legendre transformation. Furthermore, multisympltetic structure can also be obtained. A concrete example can be found in

Chen and Liu, Applied Mathematics and Computation 204 (2008) 58–62

You can find enormous examples in the literature. First find the appropriate Lagrangian then the momenta related with your PDE and construct the Hamiltonian.

 Or you can try the book of Holm et al.:

 https://global.oup.com/academic/product/geometric-mechanics-and-symmetry-9780199212910?cc=us&lang=en&