A continuous, path-independent, non-conservative vector field

Question

I am trying to define a non-conservative vector field on $\mathbb{R}^2$ whose line integrals don't depend on its paths. I think it could be $$ F(x,y) = \begin{cases} (x+y, x-y) && \mbox{if $(x,y) \neq (0,0)$} \\ (2,2) &&\mbox{if $(x,y) = (0,0).$}\end{cases} $$ Is it right? Now, what about a continuous vector field with this property (non-conservative and path-independent)? Can you give me an example of it?