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  $$g=5dx\otimes dx-dx\otimes dy -dy\otimes dx +5dy\otimes dy$$

For  this  metric we have $X=grad_g f$  where $f(x,y)=-1/2(49x^2-10xy+y^2)$
The  motivation for  consideration of  a  big number as $10$ in $X$ is  the  following:

**Note:** If we choose  the vector field  $$X=(y-\epsilon x)\partial_x-x\partial_y$$ where  $|\epsilon|<2$ then there is  no  an  analytic  Riemannian  metric $g$ with  $X=\text{grad}_g$. Because the orbits tend to the  singularity at the origin, spirally. This means  that the  singularity is  a  focus singularity. On the  other  hand  the  Riemannian  version of the  [Thom  gradient  conjecture](https://en.wikipedia.org/wiki/Gradient_conjecture) says  that if  an  orbit of  an analytic gradient vector  field tends to  an isolated singularity then it must approach to  the  singularity  in a  specific direction. Of  course a  focus singularity violate this condition.


**Shahshahani Gradient:**
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)