A continuum limit of the Toda lattice and the equilibrium for the constrained energy problem in the presence of an external field

We present the derivation of equations for the interval of equilibrium for the constrained energy problem in the presence of an external field and show that for some time dependence of the external field $Q(y,t)=Q(y,0)-yt/2$ the endpoints of the interval of equilibrium $[\alpha(x,t),\beta(x,t)]$ are the solutions of Continuum limit of the Toda lattice $\partial\alpha/\partial t=(\beta-\alpha)/4\,\partial\alpha/\partial x$, $\partial\beta/\partial t=(\beta-\alpha)/4\,\partial\beta/\partial x$.
Also here we investigate the continuity of the set of equilibrium in general case.