A patient presents with diplopia with limitations of adduction in the left eye and abducting saccade in the right eye. Convergence is preserved. Most likely etiology is-
This patient has:-

Limitations of adduction in left eye
Abduction (abducting saccade) in right eye
Convergence is preserved

This is seen in internuclear ophthalmoplegia.
During right horizontal (lateral) gaze the central pathway is:- 
Left frontal eye field → contralateral (right) PPRF → Right abducens nuclei (causing abduction of right eye) → Right medial longitudinal fasciculus → contralateral oculomoter nerve → medial rectus (adduction of left eye). So, abduction of right eye is accompanied by adduction of left eye for right lateral gaze. In right internuclear ophthalmoplegia there is damage to right medial longitudinal fasciculus. Therefore, signal transmission is normal upto abducens nucleus (abduction of right eye is possible), but it cannot be further transmitted to controlateral 'oculomotor nucleus (adduction of left eye is not possible). Thus, on attempting right lateral gaze, abduction of right eye is possible but adduction of left eye does not occur. There may be saccades in right eye during abduction.
Convergence is maintained in internuclear ophthalmoplagia as the pathway for convergence is different and does not involve medial longitudinal fasciculus. The convergence pathways run directly into the midbrain without involving MLF.
Partial 3rd nerve palsy
Internuclear ophthalmoplegia
Duane's reactionary syndrome
Absence of medial rectus muscle