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First, we need to figure out what we mean by “structure” – this can be used to refer to the corporate form of the company: Corporation, Limited Liability Company (LLC), etc. Or, it can be used to discuss ownership (equity) – who owns what percentage of the company, is part owned by a separate US or overseas corporate entity, etc. In either case, the answer will to some extent depend on what type of visa we’re seeking.
Here there is a great deal more flexibility, generally speaking. Few of the primary visas available for new companies will require a specific form. It would be very difficult to proceed as a “sole proprietorship” – where the company and individual are legally the same and there is no separate entity for the company – but almost any other form will work for an H-1B, for an E-1/E-2, for an L-1 (however a “branch” office, while legally permissible, may be problematic later in the process), or an O-1.
Here, the answer will be very specific to the type of visa. Majority equity ownership can be essential for an E-2, but almost entirely prohibited for an H-1B. In each case, there are exceptions – it is possible to get an E-2 for a non-majority owner in some situations, and not entirely impossible – at least in theory – to get an H-1B for an individual with a majority equity stake (though it would be extremely difficult). An L-1A can be a majority owner –it isn’t either required or prohibited – but majority equity ownership alters the nonimmigrant intent requirement for the L-1A and may make it difficult to apply for permanent residence later on. There is no blanket prohibition on majority equity ownership for an O-1…though the context of the business involved may mean that such ownership raises issues for the adjudication of the case.