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A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix $A\in ₙ\left(\mathbb{Z}\right)$ is ℤ-congruent to its transpose $A^{tr}$ is also discussed. An affirmative answer is given for the incidence matrices $C_I$ and the Tits matrices $C{̂}_I$ of positive one-peak posets I.