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For special kinematic configurations involving a single momentum scale, certain standard relations, originating from the Slavnov-Taylor identities of the theory, may be interpreted as ordinary differential equations for the ``kinetic term'' of the gluon propagator. The exact solutions of these equations exhibit poles at the origin, which are incompatible with the physical answer, known to diverge only logarithmically; their elimination hinges on the validity of two integral conditions that we denominate ``asymmetric'' and ``symmetric'' sum rules, depending on the kinematics employed in their derivation. The corresponding integrands contain components of the three-gluon vertex and the ghost-gluon kernel, whose dynamics are constrained when the sum rules are imposed. For the numerical treatment we single out the asymmetric sum rule, given that its support stems predominantly from low and intermediate energy regimes of the defining integral, which are physically more interesting. Adopting a combined approach based on Schwinger-Dyson equations and lattice simulations, we demonstrate how the sum rule clearly favors the suppression of an effective form factor entering in the definition of its kernel. The results of the present work offer an additional vantage point into the rich and complex structure of the three-point sector of QCD.