I'm struggling with where to start on the following question
Consider the gradient vector field
$$ \frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} \frac{\partial V}{\partial x} \\ \frac{\partial V}{\partial y} \end{pmatrix} $$ where $V(x,y)=x^3y+xy^3-4xy$. Show that $(0,0)+E_{\lambda}$ for the two eigenspaces $E_{\lambda}$ of $D^2V(0,0)$ is invariant under the flow.
I don't really know where to start. What is $D^2V(0,0)$? The Jacobi matrix?
Phase plane:
I have calculated the Jacobi matrix: $$ J(x,y)=\begin{pmatrix} 6xy && 3x^2+3y^2-4 \\ 3x^2+3y^2-4 && 6xy \end{pmatrix} $$
I found that $(0,0)$ is an equilibrium point: an (instable) saddle point. I then found the eigenvalues and eigenvectors of $J(0,0)$: $\lambda_1=4, \eta_1=(1,-1)^t, \lambda_2=-4, \eta_2=(1,1)^t$.
So then $(0,0) + E_{\lambda_1}=E_{\lambda_1}= \{(1, -1)^t\}$ and $E_{\lambda_2}=\{(1, 1)^t\}$.
Then I'm stuck, mostly because I don't know what they mean. Invariant under what flow? And is $D^2V(0,0)$ what I assumed above?