Three Mutually Adjacent Leonard Pairs
  Let (A,B) and (C,D) denote Leonard pairs on V. We say these pairs are adjacent whenever each basis for V which is standard for (A,B) (resp. (C,D)) is split for (C,D) (resp. (A,B)). Our main results are as follows:   Theorem 1. There exists at most 3 mutually adjacent Leonard pairs on V provided the dimension of V is at least 2.   Theorem 2. Let (A,B), (C,D), and (E,F) denote three mutually adjacent Leonard pairs on V. There for each of these pairs, the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression.   Theorem 3. Let (A,B) denote a Leonard pair on V whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Then there exist Leonard pairs (C,D) and (E,F) on V such that (A,B), (C,D), and (E,F) are mutually adjacent. 
Brian Hartwig