Given two smooth vector fields $X$ and $Y: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $X(x_1,x_2) = (X_1(x_1,x_2), X_2(x_1,x_2))$ and $Y(x_1,x_2) = (Y_1(x_1,x_2), Y2(x_1,x_2))$, with the Jacobian matrices J(X) and J(Y). Suppose $J(X) = J(Y)A$, where $A$ is a $2\times 2$ constant matrix,
\begin{equation} \left( \begin{array}{cc} \frac{\partial X_1}{\partial x_1} & \frac{\partial X_1}{\partial x_2}\\ \frac{\partial X_2}{\partial x_1} & \frac{\partial X_2}{\partial x_2} \end{array} \right) = \left( \begin{array}{cc} \frac{\partial Y_1}{\partial x_1} & \frac{\partial Y_1}{\partial x_2}\\ \frac{\partial Y_2}{\partial x_1} & \frac{\partial Y_2}{\partial x_2} \end{array} \right) \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}\right). \end{equation} Is it possible to express the vector field $X$ solely in terms of $Y$ and $a_{ij}$?
Do the relations between the Jacobian matrices of X and Y have any significance or implications when the matrix A belongs to some Lie Group of Matrix?