I recently ran into an odd looking ODE, that has the following form. It is the gradient of the function $w(\hat{x},\hat{E}(\hat{x}))$, where both $\hat{x}$ and $\hat{E}$ are vectors. Taking the total derivative gives the following system of ODEs:
$$\frac{d w(\hat{x},\hat{E}(\hat{x}))}{d \hat{x}} = \frac{\partial w}{\partial \hat{x}} + (\frac{\partial w}{\partial \hat{E}})^T \frac{\partial \hat{E}}{\partial \hat{x}}$$
Where the derivatives with respect to vectors are the gradients of those vectors and $\frac{\partial E}{\partial \hat{x}}$ denotes the Jacobian of the vector E.
My question is, for a system like this, what does the Jacobian of that vector E tell us about the system? I.e. what do its eigenvalues and eigenvectors tell us? It seems superficially similar to the linear system $d\hat{x}/dt = \hat{b} + A \hat{x}$, but with the partial derivatives I'm not sure how to analyze the system.