On the null-vectors in the spectra of the 2D integrable hierarchies

We propose an alternative description of the spectrum of local fields in the classical limit of the integrable quantum field theories. It is close to similar constructions used in the geometrical treatment of W-gravities . Our approach provides a systematic way of deriving the null-vectors that appear in this construction. We present explicit results for the case of the A1(1)-(m)KdV and the A2(2)-(m)KdV hierarchies, different classical limits of 2D CFTʹs. In the former case our results coincide with the classical limit of the construction . Some hints about quantization and off-critical treatment are also given.