Askey-Wilson relations and Leonard pairs

It is known that if (A,A∗) is a Leonard pair, then the linear transformations A, A∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A∗A2 − γ (AA∗+A∗A) − ̺A∗ = γ∗A2 + ωA + η I, A ∗2 A− βA ∗ AA ∗+ AA∗2− γ∗(A∗A+AA∗) − ̺∗A = γA∗2+ ωA∗+ η∗I, for some scalars β, γ, γ∗, ̺, ̺∗, ω, η, η∗. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations. AMS 2000 MSC Classification: 05E35, 33D45, 33C45.