Nowadays it is known that for a properly chosen isospectral problem

${\psi}_{x}=U\psi $, where

$U=U(u,\lambda )$ is a matrix depending on

$u=({u}_{1}(x,t),\xb7\xb7\xb7,{u}_{p}(x,t))$ and a spectral parameter

$\lambda $, we can relate it to a hierarchy of t-evolution equations

${\psi}_{{t}_{n}}={V}^{\left(n\right)}\psi $ such that the compatibility condition of the above two equations, which reads

${U}_{{t}_{n}}-{V}_{x}^{\left(n\right)}+UV-VU=0,$ represents a meaningful hierarchy of nonlinear evolution equations on u:

${u}_{{t}_{n}}={f}_{n}\left(u\right)$. It is also known that in most cases the above hierarchy of equations can be cast to their Hamiltonian form:

${u}_{{t}_{n}}={f}_{n}\left(u\right)=J\delta {H}_{n}/\delta u$, where J is a Hamiltonian operator. The main result of this paper is a formula for the Poisson bracket:

$\left\{H\right(\lambda ),H(\mu \left)\right\}=(d/dx)f(\lambda ,\mu )$, where

$H\left(\lambda \right)=\sum {H}_{n}{\lambda}^{-n}$ and the function f(

$\lambda $,

$\mu )$ is explicitly constructed. As an immediate consequence of this formula, it is shown that

$\{{H}_{m},{H}_{n}\}$ are total derivatives with respect to x. It means that the flows

$J\delta {H}_{n}/\delta u$ commute to each other, thus it proves generally that the hierarchy of equations

${u}_{{t}_{n}}=J\delta {H}_{n}/\delta u$ are Liouville integrable. Moreover in a subsequent paper we show that the explicit form of f(

$\lambda $,

$\mu )$ can be used to construct effectively the Hamiltonians for the stationary Lagrangian equations

$\delta {H}_{n}/\delta u=0$.