Smooth foliations by circles of S^7 with unbounded periods and nonlinearizable multicentres

We give an example of a (Formula presented.) vector field (Formula presented.), defined in a neighbourhood (Formula presented.) of (Formula presented.), such that (Formula presented.) is foliated by closed integral curves of (Formula presented.), the differential (Formula presented.) at (Formula presented.) defines a one-parameter group of non-degenerate rotations and (Formula presented.) is not orbitally equivalent to its linearization. Such a vector field (Formula presented.) has the first integral (Formula presented.), and its main feature is that its period function is locally unbounded near the stationary point. This proves in the (Formula presented.) category that the classical Poincaré centre theorem, true for planar non-degenerate centres, is not generalizable to multicentres. Such an example is obtained through a careful study and a suitable modification of a celebrated example by Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], by blowing up the stationary point at the origin and through the construction of a smooth one-parameter family of foliations by circles of (Formula presented.) whose orbits have unbounded lengths (equivalently, unbounded periods) for each value of the parameter and which smoothly converges to the Hopf fibration (Formula presented.).