Gauge transformation and generating operators for a quadratic bundle

A gauge-covariant formulation of the theory of the generating operator $\Lambda$ for a quadratic bundle is found. On this basis, the method of expansion with respect to “squared solutions” is applied to the auxiliary linear problem
$$ \left\{iS_0(x)\frac{d}{dx}+\lambda S_1(x)-\lambda^2\right\}\tilde v(x,\lambda)=0. $$
Thus, for nonlinear evolution equations associated with this problem a hierarchy of Hamiltonian structures is obtained and their complete integrability is proved. Some examples, including equations of Landau–Lifshitz type, are considered for suitable reduction.