Results in Quantum Field Theories, such as anomalous dimensions or scattering amplitudes are known to exhibit rich properities under the Galois coaction that acts on period integrals. I will show how these phenomena arise directly from an integrability setup describing certain anomalous dimensions of an integrable four-dimensional scalar model. I will also discuss a conjectural all-loop differential equation for these quantities, constructed from derivatives with respect to Riemann zeta values.