People in L-2 status are permitted to request employment authorization. Have your spouse's employer complete Form I-765 on your behalf. You will not be permitted to work while the application is pending if you are already no longer in H-1 status.
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You can work in L-2 status once you apply for and receive your EAD. Your employer can apply for a green card for you while you are in L-2 status.
Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
You should be eligible to stay and work so long as your spouse maintains her L-1 status. Green card application can also run parallel to these.
This response is general in nature and cannot be construed as legal advice, given that not enough facts are known. Any comments offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship. You are encouraged to seek independent and private counseling for a complete review of your case.