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The gauge-invariant,nonlocal,dynamical quark model is shown to generate minimal vertices that satisfy the Ward-Takahashi identities and the flat-like form of the light-cone pion distribution amplitudes.Non-flat form amplitudes can be produced only if we take a finite momentum cutoff and include nonzero pion mass corrections or go beyond the minimal vertices.A by-product of our investigation shows that the variable u appearing in light-cone pion distribution amplitudes is just the standard Feynman parameter in the Feynman parameter integrals.