You may first want to check with the company's immigration attorney whether you qualify for L-1A, employment abroad rule in managerial capacity, etc.
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You should speak with your attorney about these issues. As long as a change of status was filed for you along with the request for L-1A classification, there is no reason to leave the U.S. I would also talk to the company about starting your green card process before you run out of L time. Status not visa is important. You need to stop working with your H status expires if the L-A is not yet approved. If this is a danger you may want to consider premium processing so there won't be a gap in your right to work.
What did the company's immigration lawyer say when you asked him/her?
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
Have them premium process the L1A application so a decision is issued in 2-3 weeks.
Business Immigration Attorney with 17 years of immigration law experience. Call 800-688-7892 or visit www.ImmigrationDesk.com. Law Office of Anu Gupta. The advice suggested here is for general information only and not to be construed as legal advice.