Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3).