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We reconsider the contribution due to $\pi a_1$-mixing to the anomalous$\gamma\to\pi^+\pi^0\pi^-$ amplitude from the standpoint of the low-energytheorem $F^{\pi}=e f_\pi^2 F^{3\pi}$, which relates the electromagnetic formfactor $F_{\pi^0\to\gamma\gamma}=F^\pi$ with the form factor$F_{\gamma\to\pi^+\pi^0\pi^-}=F^{3\pi}$ both taken at vanishing momenta ofmesons. Our approach is based on a recently proposed covariant diagonalizationof $\pi a_1$-mixing within a standard effective QCD-inspired meson Lagrangianobtained in the framework of the Nambu-Jona-Lasinio model. We show that the twosurface terms appearing in the calculation of the anomalous triangle quarkdiagrams or AVV- and AAA-type amplitudes are uniquely fixed by this theorem. Asa result, both form factors $F^\pi$ and $F^{3\pi}$ are not affected by the $\pia_1$-mixing, but the concept of vector meson dominance (VMD) fails for$\gamma\to\pi^+\pi^0\pi^-$.

Low-energy theorem for $\gamma\to 3\pi$: surface terms against $\pi a_1$-mixing