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 Riemanni8an  metric  $$g=5dx\otimes dx-dx\otimes dy -dy\otimes dx +5dy\otimes dy$$