College Answer

(a) Horizontal gaze palsy (right)

(b) Any two:
 Frontal eye field
 Posterior hemispheric lesion
 Pre-pontine Reticular Formation (PPRF)
 Abducens (VI) nerve nucleus

(c) Right sided paralysis.

Discussion

The college loves gaze palsies. And cranial nerve lesions in general.

To answer this specific question, I resorted to the free-ish copy of "Fundamentals of Neurology: An Illustrated Guide" by Mattle and Mumenthaler, available via Google Books. Horisontal gaze palsies are dealt with on page 188.

The specific disturbance is an abducens nerve palsy - the affected eye cannot abduct past midline. Judging by the college answer, the palsy is unilateral.  Causes of a unilateral 6th nerve palsy include the following:

• Head injury (most common) with BOSF
• Raised intracranial pressure
• Localising lesion.... at any number of levels:
  • Damage to the frontal eye field of the frontal lobe, which occupies some of the middle frontal gyrus
  • Damage to the posterior hemispheres, which would be accompanied by a hemianopia
  • Brainstem (tumour, stroke) - the paramedian prepontine reticular formation mentioned in this question, which receives information from higher cortical centres and transmits them to the abducens nucleus.
  • Petrous portion of temporal bone (otitis media-associated osteomyelitis, mastoiditis)
  • Clivus (intraforaminal extension of nasopharyngeal carcinoma or similar)
  • Cavernous sinus (thrombosis)
  • Superior orbital fissue (base of skull fracture)
  • Any damn where (basal forms of meningitis, eg sarcoidosis, tuberculosis, cryptococcus)

The college then claims that the hemiplegic side is ipsilateral to the horisontal gaze palsy. This is weird. The damage to the brainstem should be at the level of the pons, and therefore the hemiplesia should be contralateral. An excellent 2013 article by Azarmina et al ("The Six Syndromes of the Sixth Cranial Nerve") discusses the various diffuse and focal ways in which the sixth nerve can be damaged. All the pyramidal signs are contralateral in these scenarios.

Abducens Nerve Palsies

CN IV: Abducens	Expected features Failure to abduct the affected eye
	Unilateral lesion: Head injury (most common) with BOSF Raised intracranial pressure Localising lesion.... at any number of levels: Damage to the frontal eye field of the frontal lobe, which occupies some of the middle frontal gyrus Damage to the posterior hemispheres, which would be accompanied by a hemianopia Brainstem (tumour, stroke) - the paramedian prepontine reticular formation mentioned in this question, which receives information from higher cortical centres and transmits them to the abducens nucleus. Petrous portion of temporal bone (otitis media-associated osteomyelitis, mastoiditis) Clivus (intraforaminal extension of nasopharyngeal carcinoma or similar) Cavernous sinus (thrombosis) Superior orbital fissue (base of skull fracture) Any damn where (basal forms of meningitis, eg sarcoidosis, tuberculosis, cryptococcus)
	Bilateral lesion:Freakishly rare Diabetes-associated neuropathy Myasthenia gravis Generally speaking, polyneuropathies such as Guillain-Barre

The table below is a fragment of the greater Table of Conjugate Gaze Palsies. It may also be helpful in localising this lesion.