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)