  The construction of a solution of the perturbed KdV equation encounters
obstacles to asymptotic integrability beyond the first order, when the
zero-order approximation is a multiple-soliton wave. In the standard analysis,
the obstacles lead to the loss of integrability of the Normal Form, resulting
in a zero-order term, which does not have the simple structure of the solution
of the unperturbed equation. Exploiting the freedom in the perturbative
expansion, we propose an algorithm that shifts the effect of the obstacles from
the Normal Form to the higher-order terms. The Normal Form then remains
integrable, and the zero-order approximation retains the multiple-soliton
structure of the unperturbed solution. The obstacles are expressed in terms of
symmetries of the unperturbed equation, and decay exponentially away from the
soliton-interaction region. As a result, they generate a bounded correction
term, which decays exponentially away from the origin in the x-t plane. The
computation is performed through second order.
