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In a previous paper, we showed that any Jacobi field along a harmonic mapfrom the 2-sphere to the complex projective plane is integrable (i.e., istangent to a smooth variation through harmonic maps). In this paper, incontrast, we show that there are (non-full) harmonic maps from the 2-sphere tothe 3-sphere and 4-sphere which have non-integrable Jacobi fields. This isparticularly surprising in the case of the 3-sphere where the space of harmonicmaps of any degree is a smooth manifold, each map having image in a totallygeodesic 2-sphere.

Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable