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Let = [0,1) be the additive group of real numbers modulo 1, α ∈ be an irrational number and t ∈ . We study ergodicity of skew product extensions T : × ℤ² → × ℤ², $T(x,s₁,s₂)=(x+\alpha ,s₁+2{\chi }_{[0,1/2)}\left(x\right)-1,s₂+2{\chi }_{[0,1/2)}(x+t)-1)$.