= Solution
\b[No.] On the round <unit sphere>[sphere], take a latitude circle other than the equator. Rotations about the polar axis preserve the circle and act transitively on its points, so all the isometries $\phi_{x,y}$ exist. The latitude separates the sphere into two caps of unequal area, so those caps are not isometric. It is not a great circle and therefore is not a geodesic; equivalently, its geodesic curvature is nonzero. This is a <homogeneous nongeodesic latitude>.
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