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We present a novel method for computing the nonperturbative kinetic term of the gluon propagator from an exactly solvable ordinary differential equation, whose origin is the fundamental Slavnov-Taylor identity satisfied by the three-gluon vertex, evaluated in a special kinematic limit. The main ingredients comprising the solution are a well-known projection of the three-gluon vertex, simulated on the lattice, and a particular derivative of the ghost-gluon kernel, whose approximate form is derived from a standard Schwinger-Dyson equation. Crucially, the physical requirement of a pole-free answer determines completely the form of the initial condition, whose value is calculated from a specific integral containing the same ingredients as the solution itself. This outstanding feature fixes uniquely, at least in principle, the form of the kinetic term, once the ingredients of the differential equation have been accurately evaluated. Furthermore, in the case where the gluon propagator has been independently accessed from the lattice, this property leads to the unambiguous extraction of the momentum-dependent effective gluon mass. The practical implementation of this method is carried out in detail, and the required approximations and theoretical assumptions are duly highlighted. The systematic improvement of this approach through the detailed computation of one of its pivotal components is briefly outlined.