Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top

We propose a perturbation algorithm for Hamiltonian systems on Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical systems. Given system preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$, when we add the subalgebra $\mathbb{B}$ will no longer preserved. show how to transform perturbed dynamical to preserve up to terms quadratic in perturbation. apply this method to study dynamics non-autonomous symmetric Rigid Body. In example our algebraic transform plays role of Iterative Lemma in proof a KAM-like statement.