On Lax operators

We define a Lax operator as monic pseudodifferential L(∂) of order N ≥ 1, such that the equations $$\frac{\partial L(\partial)}{\partial t_k}=[(L^\frac{k}{N}(\partial))_+,L(\partial)]$$ are consistent and non-zero for infinitely many positive integers k. Consistency an equation means its flow is defined by evolutionary vector field. In present paper we demonstrate traditional theory KP N-th KdV hierarchies holds arbitrary scalar operators.