In his paper "New integrable Case of Kirchoff's equations" (Theor. and Math. Physics. 2001) V.V. Sokolov proposed a new integrable case with additional integral of fourth degree. We have shown that the integral can be written in a more natural form and consider its generalization to a bundle of Poisson brackets.
Borisov A. V., Mamaev I. S., Sokolov V. V., A New Integrable Case on $so(4)$, Doklady Physics, 2001, vol. 46, no. 12, pp. 888-889