  We consider the conformal properties of geometries described by higher-rank
line elements. A crucial role is played by the conformal Killing equation
(CKE). We introduce the concept of null-flat spaces in which the line element
can be written as ${ds}^r=r!d\zeta_1\cdots d\zeta_r$. We then show that, for
null-flat spaces, the critical dimension, for which the CKE has infinitely many
solutions, is equal to the rank of the metric. Therefore, in order to construct
an integrable conformal field theory in 4 dimensions we need to rely on
fourth-rank geometry. We consider the simple model ${\cal L}={1\over 4}
G^{\mu\nu\lambda\rho}\partial_\mu\phi\partial_\nu\phi\partial_\lambda\phi
\partial_\rho\phi$ and show that it is an integrable conformal model in 4
dimensions. Furthermore, the associated symmetry group is ${Vir}^4$.
