I'd like to register or to reregister Jira for the company. Is it possible and if it possible how can I do this? Or Jira and other products register only for private person?
I'm not legal but I do think you can own your license and use it for work as long as it's fitting your company's need and if they are OK with you being the owner of the license. The only thing you can't do is use your license twice (e.g. in your private server + at work).
License have no distinction between private or commercial usage. That being said product licenses are based on server ID so if you're re-installing please make sure to associate your new server ID with your license as described here: https://community.atlassian.com/t5/Jira-questions/JIRA-License-transfer/qaq-p/18017
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