ИСТИНА

The Adler–van Moerbeke integrable case is an integrable system defined by the Euler equations on the Lie algebra so(4). The Hamiltonian of this system is quadratic, and the additional integral is a polynomial of degree four. This integrable case is connected with various interesting algebraic constructions. For example, it can be obtained with the help of the “argument shift method” applied to the Lie algebra so(4) considered as the normal form of the exceptional simple Lie algebra g2. This method for constructing complete involutive families of polynomials on Lie algebras was developed by A.S.Mischenko and A.T.Fomenko in 1978-79. Later, in 1984, M.Adler and P.van Moerbeke found this integrable case investigating left-invariant metrics on SO(4) that are algebraically completely integrable. Also, in 1986, A.G.Reyman and M.A.Semenov-Tian-Shansky found a Lax pair for this integrable system. The topological analisys of the Adler–van Moerbeke integrable case was recently done by A.A.Oshemkov, S.V.Sokolov, P.E.Ryabov. In particular, we describe its singularities, their types, and the bifurcation diagram of the corresponding momentum mapping. Note that there are several known integrable cases on so(4) with quartic additional integrals, and the obtained results show qualitative differences of the Adler–van Moerbeke integrable case from them.