The Korteweg-de Vries equation on the Uhlenbeck manifold

It is known that the KdV equation with respect to the function $p=p(x,t)$, periodic in the variable $x$, can be understood as a vector field $v(p)=-p''' + 6pp'$. It is also known that the solution $p(x,t)$ of the KdV equation and the corresponding eigenfunction $y(x,t)$ of the Schrödinger operator with the potential $p(x,t)$ are related by the equation $\dot{y} = -4y'''+ 6 p(x,t) y' + 3 p'(x,t)$. We will show that this equation can be understood as a vector field on the Karen Uhlenbeck manifold of triples $(p,\lambda,y)$ satisfying the Schrödinger equation.