D-E classification of the local extensions of $SU_2$ current algebras

A method is developed for constructing single valued rational 4-point functions of primary fields for $su_2$ conformal current algebra satisfying the Knizhnik–Zamolodchikov equation. For integer conformal dimensions $\Delta$ these rational solutions are proven to be in one-to-one correspondence with non-diagonal modular invariant partition functions of the D-even and E-even series of the ADE classification.