On conformally invariant equations on ${\bf R^n$}-II. Exponential invariance

In this paper we provide a complete characterization of fully nonlinear differential operators of any integer order on $\R^n$, which exhibit conformal invariance of exponential type. In this way we intend to complete the work that we undertook in [Li, Mastrolia, Monticelli, On conformally invariant equations on ${\bf R^n$} (2012, submitted)], where we introduced the family of elementary conformal tensors $\{T_{m,\alpha}^u\}$ in order to describe all fully nonlinear differential operators of any integer order on $\R^n$ which are conformally invariant of degree $\alpha\neq0$. Examples of the differential operators that we study in this paper are those related to the Q--curvature equation on $\R^4$ and to the Gauss equation on $\R^2$.