1. CR automorphisms

Problem 1.1.
[V. Matveev] Are there nonspherical, compact, nondegenerate CR manifolds which admit essential conformal vector fields?
Remark. For strictly pseudoconvex CR manifolds, Schoen proved that the answer is "no"

Remark. For the analogous problem in Riemannian geometry, C. Frances showed that the answer is "yes" in signature $(n2,2)$. One might start by trying to adapt his example to the CR setting.

Cite this as: AimPL: Analysis and geometry on pseudohermitian manifolds, available at http://aimpl.org/pshermitian.