For ADaaS snap-in's, the external_system_name is provided directly in the snap-in manifest and set by the snap-in developer.

It should start with a capital letter, and include only letters, numbers, underscore, period, or spaces. It has to conform to this ^[A-Z][A-Za-z0-9._ ]*[A-Za-z0-9]$ regex.

For native integrations, such as Jira, Salesforce, ... the external_system_name is set by the AirSync team.