Consider the vector field $$X=(y-10x)\partial_x-x\partial_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient vector field with respect to the Riemannian metric $$g=5dx\otimes dx-dx\otimes dy -dy\otimes dx +5dy\otimes dy$$ As an interesting example, please search "Shahshahani Gradient". This gradient corresponds to a Riemannian metric on the phase space of certain vector field which is not a gradient system with respect to the standard metric but is a gradient vector field with respect to the Shahshahani Riemannian metric. See here for some explanation on [Shahshahani metric.](https://books.google.com/books?id=AdKnOpCsH3gC&pg=PA52&lpg=PA52&dq=Shahshahani++Gradient++gradient+vector++field&source=bl&ots=huTRLFy49f&sig=LKs-EYCuLjnlK5t9KtOO18R2_fA&hl=en&sa=X&ved=0ahUKEwjZkOu_8bbbAhW0yqYKHUltC-g4ChDoAQgrMAI#v=onepage&q=Shahshahani%20%20Gradient%20%20gradient%20vector%20%20field&f=false)