Final answer:
The protozoan capable of causing both brain cysts and corneal infections, as referenced in the student's question, is a species of Acanthamoeba.
Step-by-step explanation:
The free-living protozoan seen in a cyst formed within brain tissue and also capable of causing corneal infection is a species of Acanthamoeba. The genus Acanthamoeba contains parasitic species that can lead to infections in various parts of the body, including the eyes and central nervous system. For instance, Acanthamoeba keratitis is an infection of the eye that can result from activities such as wearing improperly disinfected contact lenses or swimming while wearing them. This condition can cause significant damage to the cornea and, if left untreated, may lead to severe impairment of vision or even blindness. Additionally, these protozoans are known to cause granulomatous amoebic encephalitis (GAE), which is nearly always fatal.