  We consider the sphere $\Sph^{2n+1}$ equipped with its standard CR structure.
In this paper we construct explicit contact forms on $\Sph^{2n+1}\setminus
\Sph^{2k+1}$, which are conformal to the standard one and whose related Webster
metrics have constant Webster curvature; in particular the curvature is
positive if $2k< n-2$. As main applications, we provide two perturbative
results. In the first one we prove the existence of infinitely many contact
structures on $\Sph^{2n+1}\setminus \tau(\Sph^{1})$ conformal to the standard
one and having constant Webster curvature, where $\tau(\Sph^{1})$ is a small
perturbation of $\Sph^1$. In the second application, we show that there exist
infinitely many bifurcating branches of periodic solutions to the CR Yamabe
problem on $\Sph^{2n+1}\setminus \Sph^{1}$ having constant Webster curvature.
