On rational integrability of Euler equations on Lie algebra \(\mathrm{so}(4, \mathbb C)\)
The authors of this very interesting paper study a class of Euler equations on the complex Lie algebra \(L\), that is \(\text{so}(4,\mathbb C)\) with a diagonal quadratic Hamiltonian \(H\). The system of these equations has the form \[ {dx_i\over dt}=\{x_i,H\}, \quad 1\leq i\leq n, \] where the element \(x=\sum_{i=1}^{n}x_ie_i^{\ast }\in L^{\ast }\) (\(L^{\ast }\) is dual to \(L\)), \(x_i\in C^{\infty }(L^{\ast })\) (\(1\leq i\leq n\)), \(\{e_1^{\ast },e_2^{\ast },\dots,e_n^{\ast }\}\) is the basis dual to a fixed basis \(\{e_1,e_2,\dots,e_n\}\) of \(L\), \(\{x_i,H\}\) is the Lie-Poisson bracket, and \(H\in C^{\infty }(L^{\ast })\) is a given function, that is a diagonal quadratic Hamiltonian. It is well known that the above stated system always admits three functionally independent first integrals in polynomial form. The authors prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. The fourth integral is obtained in explicit form.
scientific article; zbMATH DE number 5279161
scientific article