  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.
