Parameterized norm form equations with arithmetic progressions

Let α be a zero of the Thomas polynomial X3−(a−1)X2−(a+2)X−1. We find all algebraic numbers μ=x0+x1α+x2α2∈Z[α], such that x0,x1,x2∈Z forms an arithmetic progression and the norm of μ is less than |2a+1|. In order to find all progressions we reduce our problem to solve a family of Thue equations and solve this family completely.