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We study invariant contact $p$-spheres on principal circle-bundles and solve the corresponding existence problem in dimension 3. Moreover, we show that contact $p$- spheres can only exist on $(4n-1)$-dimensional manifolds and we construct examples of contact $p$-spheres on such manifolds. We also consider relations between tautness and roundness, a regularity property concerning the Reeb vector fields of the contact forms in a contact $p$-sphere.