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For the E-1 visa, is it possible for me (owner of the company) to apply for the visa for myself (as an employee of my own company)?

The E visa is one of the few visas that allows foreign nationals to start up their own companies in the U.S. Along those lines, therefore, the answer is, yes, you apply for the visa yourself as the E1 treaty trader. However, the E1 visa has a number of requirements that go along with it, so you should be sure to meet those requirements.