The Dunkl operators and integrability

In the talk I'll explain which analogs of the classical one-dimensional Dunkl operator can be intertwinned in the Cherednik algebra with an operator of differentiation or integration. The corollaries will be discussed. The key notions are the following.
A reflection in $\mathbb R$: $s[f](x)=f(-x)$;
The Dunkl operator in $\mathbb R$: $\nabla=\frac d{dx}-\frac{k}{x}s$, where $k$- is a non-negative integer.
An analog of Dunkl operator on $\mathbb R$: $\nabla_{\omega}=\frac d{dx}-(\ln |\omega(x)|)'s$, where $\omega$ is an even function;
The Cherednik algebra: $A=\langle 1, x, d/dx, s \rangle$;
$V$ interwins $\nabla$ and $\frac d{dx}$, if $\nabla\circ V=V\circ\frac d{dx}$.