f_{\l,\mu} >> 
{{\mathcal{D}}}_{\l,\mu}^2 (M) \strut
\end{CD}
\label{TheDiagram}$$

For all $\d \not=\frac{2}{n},\frac{n+2}{2n},1,\frac{n+1}{n}, 
\frac{n+2}{n},$ the deformation of the space of symbols ${\cal S}_{\d,2}(M)$ by the space ${{\mathcal{D}}}_{\l,\mu}^2(M)$ as a ${\mathrm{Diff}}(M)$-module is given as follows: for all $P=P^{ij}\xi_i \xi_j +P^i\xi_i + P_0\in {\cal S}_{\d,2}(M),$ one has $$\bar f_{\d}\cdot (P  )=
T^{ij}\xi_i\xi_j+T^{i}\xi_i+T^{0},$$ where $$\begin{aligned}
T^{ij}&=&
(f_{\d}\,P)^{ij},
\nonumber\\\label{ExplAction}
T^i&=&
(f_{\d}\,P)^i\;+\;
\displaystyle
\frac{n(\mu+\l -1)}{(2+n(1-\d ))(2-n\delta)}\,{\cal A}^i_{kl}(f^{-1})
(f_{\d}\,P)^{kl},\\[2mm]
\displaystyle
T^0&=&
(f_\d\, P)_0
\;-\;
\displaystyle
\frac{n \l(\mu -1)}{(2+n(1-2\d))(1-\d )(1+n(1-\d))}\,{\cal B}_{kl}(f^{-1})
(f_{\d}\, P)^{kl},\nonumber\end{aligned}$$ and $f_\d$ is the action (\[actsym\]). \[MainAct\]

[**Proof.**]{} The proof is a simple computation using (\[comp\]) and the formul[æ]{} $$\begin{aligned}
\n_i\n_j \,{f^*_{\d}}^{-1} P^{kl}&=&{f^*_{\d}}^{-1} \n_i \n_j\, P^{kl} - 
\mathrm{Sym}_{l,k}\left (
{f^*_{\d}}^{-1} \n_i\, P^{tl} \;\ell(f)_{tj}^k \right) + 
{f^*_{\d}}^{-1} \n_u\, P^{kl} \; \ell(f)_{ij}^u \nonumber \\
&& +\d \left ({f^*_{\d}}^{-1} \n_i\, P^{kl} \;\ell(f)_{j}\right) 
-\mathrm{Sym}_{k,l} \left( \n_j \ell(f)^k_{it}\; {f^*_{\d}}^{-1} 
P^{tl}\right) \\
&&+\d \n_j \ell(f)_i \; {f^*_{\d}}^{-1} P^{kl} 
-\mathrm{Sym}_{k,l}\left( \ell(f)_{it}^k \n_j \, {f^*_{\d}}^{-1} 
P^{tl}\right) +\d \ell(f)_i\,\n_j {f^*_{\d}}^{-1}P^{kl}
\nonumber\\[2mm]
\n_{i} \, f^* \phi&=& f^* \n_i \phi +\l \,\ell(f^{-1})_{i}\,  f^* \phi
\nonumber\\[2mm]
\n_j \n_{i} \, f^* \phi&=& f^* \n_j \n_i \phi +\ell(f^{-1})_{ji}^t \, f^* 
\n_t \phi +\l \mathrm{Sym}_{i,j} \,  \ell(f^{-1})_j\, f^* \n_i \phi 
\nonumber\\[2mm]
&& +\left (\l \n_j\, \ell(f^{-1})_{i}
+ \l^2 \ell(f^{-1})_{j}\, \ell(f^{-1})_{i}\right) \, f^* \phi
\nonumber\\[2mm]
f^* R_{jk}-R_{jk}&=& \n_i\, \ell(f^{-1})^i_{jk}-\n_j \, \ell(f^{-1})_k 
- 