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We obtain that Sarnak's M\"{o}bius Disjointness Conjecture holds for productflows between affine linear flows on compact abelian groups of zero topologicalentropy and a class of rigid dynamical systems. To prove this, we show anestimate for the average value of the product of the M\"obius function and anypolynomial phase over short intervals and arithmetic progressionssimultaneously. In addition, we prove that the logarithmically averagedM\"obius Disjointness Conjecture holds for the product flow between any affinelinear flow on a compact abelian group of zero entropy and any rigid dynamicalsystem. As an application, we show that the logarithmically averaged M\"obiusDisjointness Conjecture holds for every Lipschitz continuous skew productdynamical system on $\mathbb{T}^2$ over a rotation of the circle.

M\"{o}bius Disjointness for product flows of rigid dynamical systems and affine linear flows