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We consider the anti-plane shear of an elastic solid whose boundary is partially reinforced by a thin solid film represented by the union of a finite number of open curves. The solvability of the resulting boundary value problems is complicated by the presence of end-point conditions which must be satisfied at the ends of each section of the reinforcing film. In order to avoid complicated solvability conditions which carry no clear physical meaning, we modify the boundary integral equation method using an equivalent (lower-order) reinforcement condition which leads to the desired solvability results for the corresponding boundary value problems.