  This paper is devoted to the existence of contact forms of prescribed Webster
scalar curvature on a $3-$dimensional CR compact manifold locally conformally
CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to
Kazdan-Warner type obstructions, conditions on the function $H$ to be realized
as a Webster scalar curvature have to be given. We prove new existence results
based on a new type of Euler-Hopf type formula. Our argument gives an upper
bound on the Morse index of the obtained solution. We also give a lower bound
on the number of conformal contact forms having the same Webster scalar
curvature.
