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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 Webstermetrics have constant Webster curvature; in particular the curvature ispositive if $2k< n-2$. As main applications, we provide two perturbativeresults. In the first one we prove the existence of infinitely many contactstructures on $\Sph^{2n+1}\setminus \tau(\Sph^{1})$ conformal to the standardone and having constant Webster curvature, where $\tau(\Sph^{1})$ is a smallperturbation of $\Sph^1$. In the second application, we show that there existinfinitely many bifurcating branches of periodic solutions to the CR Yamabeproblem on $\Sph^{2n+1}\setminus \Sph^{1}$ having constant Webster curvature.

Singular CR structures of constant Webster curvature and applications