With the following energy $E = \frac{1}{1 + x^2} (s - xy)^2$, where s is a constant and x, y are two variables. The dynamics of gradient descent on this energy are
$\dot{y} = -\frac{\partial E}{\partial y} = \frac{1}{1 + x^2} (s - xy) x $
and
$\dot{x} = -\frac{\partial E}{\partial x} = \frac{1}{(1 + x^2)^2} (s - xy) (y+xs) $
Clearly the fixed points are at $xy = s$, but I would like to have a closed form expression for the time evolution of this system. Are there any initial conditions of x and y such that there is an closed form expression?