Actually, currently there is no way to upload your own icons, just the system ones.
However, we already have a feature request in order to allow this in the feature which is tracked on the issue below:
I would advise you to add your comments to this feature, and also vote for this implementation.
For more information about how Atlassian implements new features, please, check the document below:
...be more productive while being fun to use at the same time. For some, getting started can be a bit intimidating. This is especially true if Jira Service Desk is your first exposure to Atlassian...
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