All solutions to Thomas’ family of Thue equations over imaginary quadratic number fields

We completely solve the family of relative Thue equations x3−(t−1)x2y−(t+2)xy2−y3=μ,x3−(t−1)x2y−(t+2)xy2−y3=μ, where the parameter tt, the root of unity μμ and the solutions xx and yy are integers in the same imaginary quadratic number field. This is achieved using the hypergeometric method for |t|≥53|t|≥53 and Baker’s method combined with a computer search using continued fractions for the remaining values of tt.