In this paper, we combine the general tools developed in  with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss , Host and Kra  and Ziegler  to study the averages (Formula presented.) associated to a triple of directions p1, p2, p3 ∈ ℤ2 that lie in general position along with 0 ∈ ℤ2. We show how to construct a “pleasant” extension of an initiallygiven ℤ2-system for which these averages admit characteristic factors with a very concrete description, involving the same structure as for those in  together with two-step pro-nilsystems (reminiscent of  and its predecessors). We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages (Formula presented.) associated to two commuting transformations T1, T2.
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