  We provide a new natural interpretation of the Lax representation for an
integrable system; that is, the spectral problem is the linearized form of a
Miura transformation between the original system and a modified version of it.
On the basis of this interpretation, we formulate a systematic method of
identifying modified integrable systems that can be mapped to a given
integrable system by Miura transformations. Thus, this method can be used to
generate new integrable systems from known systems through inverse Miura maps;
it can be applied to both continuous and discrete systems in 1+1 dimensions as
well as in 2+1 dimensions. The effectiveness of the method is illustrated using
examples such as the nonlinear Schroedinger (NLS) system, the Zakharov-Ito
system (two-component KdV), the three-wave interaction system, the
Yajima-Oikawa system, the Ablowitz-Ladik lattice (integrable space-discrete
NLS), and two (2+1)-dimensional NLS systems.
