Classification of λ-homomorphic braces on Z²

If A=(A,⊕,) is a λ-homomorphic brace with (A,⊕)=Z², then the operations in this brace are given by formulas {align*}{pmatrix}a_1_2{pmatrix}⊕{pmatrix}b_1_2{pmatrix}={pmatrix}a_1+b_1_2+b_2{pmatrix},&&{pmatrix}a_1_2{pmatrix}{pmatrix}b_1_2{pmatrix}={pmatrix}a_1_2{pmatrix}+φa_1ψa_2{pmatrix}b_1_2{pmatrix}, {align*} where φ,ψ∈ GL₂(Z) are cpecific matrices which depend on A. Not every pair (φ,ψ) lead to a brace. In the present paper we find all possible pairs (φ,ψ) of matrices from GL₂(Z) which lead to λ-homomorphic braces with (A,⊕)=Z². The obtained result gives the full classification of λ-homomorphic braces on Z² which was started by Bardakov, Neshchadim and Yadav in [J. Pure App. Algebra, V. 226, N. 6, 2022, 106961].