quadratics (cf. [@ki; @s]).

\(ii) Its kernel is the group of projective transformations ${\mathrm{PSL}}_2({\mathbb{R}})$.\
The aim of this paper is to propose a new approach for constructing the multi-dimensional conformal Schwarzian derivative. This approach was recently used in [@b2; @bo2] to introduce the multi-dimensional “projective” Schwarzian derivative. The starting point of our approach is the relation between the Schwarzian derivative (\[clas\]) and the space of Sturm-Liouville operators (see, e.g., [@wi]). The space of Sturm-Liouville operators is not isomorphic as a ${\mathrm{Diff}}(S^1)$-module to the space of differential quadratics. More precisely, the space of Sturm-Liouville operators is a non-trivial deformation of the space of differential quadratics in the sense of Neijenhuis and Richardson’s theory of deformation (see [@nr]), generated by the 1-cocycle (\[clas\]) (see [@ga] for more details). From this point of view, the multi-dimensional Schwarzian derivative is closely related to the modules of linear differential operators. To set out our approach, let us introduce some notation.\
Let $M$ be a smooth manifold. We consider the space of linear differential operators with arguments that are $\l$-densities on $M$ and values that are $\mu$-densities on $M.$ We have, therefore, a two parameter family of ${\mathrm{Diff}}(M)$-modules denoted by ${{\mathcal{D}}}_{\l,\mu}(M).$ The corresponding space of symbols 