Integrable equations and recursion operators related to the affine Lie algebras $A^{(1)}_{r}$

We have derived a family of equations related to the untwisted affine Lie algebras $A^{(1)}_{r}$ using a Coxeter $mathbb{Z}_{r+1}$ reduction. They represent the third member of the hierarchy of soliton equations related to the algebra. We also give some particular examples and impose additional reductions.