The answer may be in Thomas Lachand-Robert and Edouard Oudet

Jay P Fillmore, Bodies Symmetries of surfaces of constant width, J Differential Geometry 5 (1969) 103-110, says: the curve $$x_1=h\cos\theta-{dh\over d\theta}\sin\theta,\qquad x_2=h\sin\theta+{dh\over d\theta}\cos\theta$$ where $h=a+b\cos3\theta$, $0\lt8b\lt a$, is analytic and of constant width. If we rotate this curve in arbitrary dimensionEuclidean space ${\bf E}^n$ about an $(n-2)$-dimensional axis perpendicular to the line $\theta=0$, we obtain an analytic surface, not a sphere, of constant width in ${\bf E}^n$.

The paper may be available at http://www.lama.univ-savoie.fr/~lachand/pdfs/spheroforms.pdfhttp://www.intlpress.com/JDG/archive/1969/3-1&2-103.pdf

The answer may be in Thomas Lachand-Robert and Edouard Oudet, Bodies of constant width in arbitrary dimension, available at http://www.lama.univ-savoie.fr/~lachand/pdfs/spheroforms.pdf