(\[conf-constr-KD-full\]) we know that in the ${\overline{\rm MS}}$ scheme its off-diagonal part is induced by $\hat{\gamma}^c + 2\frac{\beta}{g}\hat{b}$. Therefore, the operator $$\begin{aligned}
\hat{B} =
{\hat{1} \over \hat{1} + {\cal J}
\left(\hat{\gamma}^c + 2 \frac{\beta}{g}\hat{b}\right)}\end{aligned}$$ diagonalize the anomalous dimension matrix[^6]. This formula can be understood as a resummation prescription for the conformal anomalies that induce the off-diagonal part. Knowing this matrix one can construct the eigenfunctions $\phi_l (x | \alpha_s) \equiv {^{QQ}\phi}_l (x, 1 | \alpha_s)$ of the ER-BL kernel [@Mue94] $$\phi_l(x | \alpha_s)=
\sum_{j=l}^\infty \frac{ x \bar x }{N_j(\frac{3}{2})}
C_j^{\frac{3}{2}}( 2 x - 1 ) B_{jl}(\alpha_s).$$ In Ref. [@Mue94] a differential equation for the eigenfunctions was derived in the conformal limit. Repeating the steps, it is easy to restore the neglected $\beta$ term. Employing the relation (see Appendix in [@Mue97]) $$\begin{aligned}
a(j, l) B_{jl}(\alpha_s)
= - \left\{\left(\hat{\gamma}^c(l)
+ 2 \frac{\beta}{g}\hat{b}(l)\right)\hat{B}\right\}_{jl}\end{aligned}$$ and the eigenvalue equation for the Gegenbauer polynomials [@BE53_2] we are able to derive an integro-differential equation for the eigenfunctions in the full theory, valid for the ${\overline{\rm MS}}$ scheme: $$\label{MSequation}
x \bar x \frac{d^2}{dx^2} \phi_l(x | \alpha_s)
+ ( l + 1 )( l + 2 ) \phi_l(x | \alpha_s)=
\frac{1}{2} \int_0^1 dy\,
\left(
\gamma_l^c(x,y,\alpha_s) + 2 \frac{\beta}{g} b_l(x,y)
