Rigidity problem of autodual clones

The rigidity problem for sets of autodual clones leads to two specific problems in clone theory: (i) What are the sets R of permutations of k={0, 1,..., k-1} such that 1) each r∈R has all cycles of the same prime length and 2) the only unary function on k which commutes with all elements of R is the identity e ? () If a clone C is the inter section of the autodual clones Pol r^□, r∈R and if it holds C⁽¹⁾={e}, what is the least n such that C⁽ⁿ⁾⊂∠J where J is the set of all projections ? We give some partial answers to these problems