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We will give an example of a smooth free action of $S^1=U(1)$ on $S^7$ whoseorbits have unbounded lenghts (equivalently: unbounded periods). As anapplication of this example we construct a $C^{\infty}$ vector field $X$,defined in a neighbourhood $U$ of $0 \in \mathbb{R}^8$, such that: $U-\{0\}$ isfoliated by closed integral curves of $X$, the differential $DX(0)$ at $0$defines a $1$-parameter group of nondegenerate rotations and $X$ is {\it not}orbitally equivalent to its linearization. This proves in the $C^{\infty}$category that the classical Poincar\'e Centre Theorem, true for planarnondegenerate centres, is not generalizable to multicentres.

Smooth circle actions on $S^7$ with unbounded periods and non-linearizable multicentres