Several cranial nerves work together to control eye movement. The main nerves responsible are:
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Oculomotor Nerve (Cranial Nerve III): This nerve plays the most significant role, controlling four out of the six muscles that move each eye. As stated by Texas Children's Hospital, "Cranial nerve 3, also called the oculomotor nerve, has the biggest job of the nerves that control eye movement. It controls 4 of the 6 eye muscles in each eye:". The Cleveland Clinic explains that this nerve "is one of the main nerves you use to control how your eyes move" and connects to muscles on multiple sides of the eyeballs. (https://my.clevelandclinic.org/health/body/21708-oculomotor-nerve)
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Trochlear Nerve (Cranial Nerve IV): This nerve controls the superior oblique muscle, enabling downward and inward eye movement. As noted by the Cleveland Clinic, "It enables movement in your eye's superior oblique muscle. This makes it possible to look down." Further, "the trochlear nerve (CNIV), also originating in midbrain, innervates the contralateral superior oblique, enabling the eye to point down while it is pointed…".
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Abducens Nerve (Cranial Nerve VI): This nerve controls the lateral rectus muscle, responsible for moving the eye outward. Damage to any of these three nerves (III, IV, or VI) will impair normal eye movement, as indicated by the MSD Manuals. (https://www.msdmanuals.com/home/brain-spinal-cord-and-nerve-disorders/cranial-nerve-disorders/overview-of-the-cranial-nerves) Johns Hopkins Medicine also highlights the role of the abducens nerve in eye movement. (https://www.hopkinsmedicine.org/health/conditions-and-diseases/multiple-cranial-neuropathies)
In summary, while the oculomotor nerve is the primary controller, the trochlear and abducens nerves are crucial for complete eye movement control. The Moebius Syndrome website succinctly states that cranial nerve III "Controls eye movement, pupil dilation, and pupillary constriction". (https://moebiussyndrome.org/wp-content/uploads/2015/07/Twelve_Cranial_Nerves.pdf)