The adjoint operators as elliptic operators Edit:
It seems  that the  link  "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains  a  talk by Loic Teyssier about  homological equations  and vanishing cycles  is  temporally  inactive.

I  asked this  question at  MSE but  I  did  not  get any  answer  so  I  ask  it  here  at  MO:
Assume  that  $M$ is  a smooth  $n$  dimensional manifold  with  $n>1$.
Is there a  lie  algebra  structure  on $\chi^{\infty} (M)$, the  space  of  all  smooth  vector  fields  on $M$,  such that   we  have the   following  property:

For  every  vector  field  $X\in \chi^{\infty}(M)$ the   operator $ad_X:\chi^{\infty}(M) \to \chi^{\infty}(M)$  with $ad_{X}(Y)=[X,Y]$ is  an  elliptic  differential operator of  positive  order (non zero order) when we restrict it to  non singular points of  $X$.

A  motivation for  this  question is the  consideration of  the  adjoint  operator by  Loic  Teyssier  in the  following  talk. But I do not  know if   this  talk imply that there  is  a  relation between the  number  of  limit  cycles  and the  adjoint operator?

Is there a pre print or  published paper extracting from   this talk?Is there 
   a published conference proceeding for the  corresponding  program in Toulouse?  What  is  the  role of the adjoint  operator in investigation of  the  number  of  limit  cycles? 

This  talk is  indicated here but it seems that it is  inactive now.
https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt
My  initial  motivation for  consideration  of  diff. operators associated with  a  vector  field  is  the  following  note:
https://arxiv.org/abs/math/0408037
The  note  has  a  false  part: " It  is  claimed that the  codimension of  the  range  of  derivation operator  is  equal  to  the  number  of  limit  cycles"  but the true  version is that "This  codimension is an upper  bound  for  the  number  of limit  cycles". The true  part  of  the "Proof" of  this  note  is  that :"around  a  hyperbolic  limit  cycle, one  can  solve  the  PDE  $X.f=g$ provided the integral of  $g$  along the  limit  cycle be  equal to zero.  Another  true  part  of  the  note is  included  in Remark 1, which actually contains  a  proof  of the  fact that the  codimension of  $D_X$ is  an  upper  bound  for  the  number  of  limit  cycles"
But  after  13 years, I think  that  my dream of  finite  codimensionality is  going to  be  collapsed. The  reason is  the following  interesting  comment  by  Lukas Geyer  
https://math.stackexchange.com/questions/1163800/elliptic-and-fredholm-partial-differential-operators
I  understand that  my   old dream is  collapsed, because perhaps his argument, in the  above  MSE link,  for  two singularities  can  be  repeated  for coexistence of one  singularity  and  one  limit  cycle.  Assume  that a  limit  cycle  $\gamma$ surrounds  a  singularity.  Consider  the space  of  all smooth (or    analytic) functions  vanishing at  $\gamma$ and  singularity.  Then  perhaps in this function space  we can separate  all  orbits  in the  interior  of  $\gamma$  by $\int_{-\infty}^{+\infty} g(\phi_{t})(x)dt$ hence  we  have  infinite  codimension.
So  I  search  for  some other  diff. operators  associated  with  a  vector  field whose some operator theoretical quantities  have some dinamical interpretation. For example  fredholm index as  a quantity  which interprate the number of attractors. This  is  a  motivation to  ask  for some  elliptic  operator in the  form  of $ad_X$  for  some  other  Lie  structures  on  $\chi^{\infty}(\mathbb{R}^2)$.
Note:
The  codimension of the  range of  differential operators for  algebraic vector field is  introduced here
Codimension of the range of certain linear operators
 A: Consider the special case $M=\mathbb R^2$ and $X\equiv(1,0)$.
Now, given any vector field $Y$ in terms of component functions as $Y(x_1,x_2)=(y_1(x_1,x_2),y_2(x_1,x_2))$, a simple calculation gives
$$
ad_XY
=
(\partial_1y_1,\partial_1y_2).
$$
This means that $ad_X=\partial_1\otimes I_2$.
Here $I_2:\mathbb R^2\to\mathbb R^2$ is the identity operator.
(In the Euclidean space we can differentiate somewhat carelessly and freely identify tangent spaces with their duals.)
Given any $x\in\mathbb R^2$ and $\xi\in T_xM$, the principal (and full) symbol is
$$
\sigma_{ad_X}(x,\xi)
=
i\xi_1\otimes I_2
:
\mathbb R^2\to\mathbb R^2.
$$
This is invertible if and only if $\xi_1\neq0$.
In particular, at any point $x$ there are non-zero tangent vectors $\xi$ (e.g. $(0,1)$) for which the principal symbol is not invertible (it vanishes entirely!).
Therefore the operator is not elliptic in the sense of invertible principal symbols.
This special case is actually not special.
The same argument works in any $\mathbb R^n$ or an open subset.
If $X$ is a non-vanishing vector field in an open subset of a manifold, you can set up local coordinates so that $X$ becomes a constant vector and the same argument goes through.
A: Any Lie bracket structure that is a first order differential operator is, by bilinearity and skew-symmetry, of the form
$$
\mathrm{ad}_X Y = A^{kp}_{ij}(X^i\partial_pY^j- Y^i\partial_pX^j)e_k,
$$
where $e_1, \dots, e_n$ is the standard basis of $\mathbb{R}^n$ and we can assume, by skew-symmetry, that $A^{kp}_{ij} = -A^{kp}_{ji}$. For example, the standard Lie bracket is when
$$A^{kp}_{ij} = \frac{1}{2}(\delta^i_p\delta^j_k - \delta^j_p\delta^i_k)$$
The symbol of the general adjoint operator is
$$
\sigma_X(\xi)Y = A^{kp}_{ij}X^i\xi_pY^je_k = \frac{1}{2}A^{kp}_{ij}\xi_p(X^iY^j-X^jY^i)e_k
$$
The vector $Y = X$ always lies in the kernel of this symbol.
