a)    Lateral part of medulla (Lateral Medullary Syndrome acceptable)
b)    Occlusion of PICA branch

Discussion

Important features here are:

• History of nausea
• Clearly some sort of serious swallowing dysfunction
• Left sided miosis
• Left sided incoordination (obviously cerebellar)
• Left sided nystagmus

So, to localise this, it is clear something must be going on at the level of the cerebellum. However, an isolated cerebellar lesion would not be producing that miosis. That would require a lesion which interrupts descending sympathetic fibres, leaving behind the parasympathetic stimulus from the midbrain. The combination of the two leads you to the lateral medulla. 

In summary, the features of lateral medullary syndrome are:

• On the side of the lesion:
  • Facial sensory loss
  • Nystagmus
  • Horner's syndrome
  • Loss of gag reflex
  • Ipsilateral ataxia with a tendency to fall to the ipsilateral side
• On the contralateral side:
  • Pain and temperature sensory loss in the extremities
• Generally:
  • Vertigo
  • Nausea
  • Dysphagia

The most likely culprit is the PICA, but a lateral medullary syndrome can also develop due to a large number of different lesions, likely accounting for the wide variation in individual brainstem vascularity. Anatomically, the damaged territory is supplied by the PICA, but PICA-specific occlusions are  apparently in the minority; about 80% of lateral medullary strokes happen because the vertebral artery is occluded (Kim et al, 2016)