Subelliptic biharmonic maps

We study subelliptic biharmonic maps i.e. smooth maps $\phi : M \to N$ from a compact strictly pseudoconvex CR manifold $M$ into a Riemannian manifold $N$ which are critical points of the energy functional $E_{2,b} (\phi ) = \frac{1}{2} \int_M \| \tau_b (\phi ) \|^2 \;\theta \wedge (d \theta )^n$. We show that $\phi : M \to N$ is a sublelliptic biharmonic map if and only if its vertical lift $\phi \circ \pi : C(M) \to N$ to the (total space of the) canonical circle bundle $S^1 \to C(M) \stackrel{\pi}{\longrightarrow} M$ is a biharmonic map with respect to the Fefferman metric $F_\theta$ on $C(M)$.