Hello, my name is ***** ***** I will be the immigration attorney helping you today.
Well yes unfortunately you only have the J1 visa through a specific sponsor. You cannot work at other jobs if you don't have a visa to work in that specific job. If you do not perform the job from your J1 visa, then the sponsor can terminate you and you would no longer have the visa. If you quit you also would no longer have the job. You could definitely be deported if you are terminated and no longer have J1 status.
I'm sorry, I wish I had better news for you. I hope you can understand that I do not make the laws and I cannot change them. I am only being honest with you.
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