Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a vector field given by $f_i(x)=\sum_{k=1}^n \sin(x_i-x_k), \forall x\in \mathbb{R}^n $. Now consider the ODE $\dot x=f(x)$. We observe that the Jacobian of $f(x)$ is singular (each row sums to zero). The intuition behind this singularity is that the vector field is invariant under shift, i.e., if $f(x)=f(x+a\mathbf{1}^n)$, where $\mathbf{1}^n$ is a vector of all ones and $a$ is any constant. What can we say about the center manifold of this ODE? Is it true that the center manifold in this case is unique and is equal to the one-dimensional subspace (i.e., the null space of the Jacobian)?
Another question is that how can I handle this kind of singularity? I am asking because I cannot apply many theorems (like the inverse function theorem) because of this singularity.
Thank you, in advance, for your comments.