Ok, perfect! You do not have to go back home to Russia! You just have to file the following forms to get Lawful Permanent Residency after you marry: an I-485
(application for Lawful Permanent Residency), I-765 (Application for work permit), I-131 (Application for Advance Parole), G-325a (Biographic data - one for each of you), and an I-693 (Medical exam that a certified doctor must fill).
You will need to file each form with supporting evidence and appropriate filing fees. You can find these forms at www.uscis.gov/forms
The supporting evidence that you would file would be birth certificates, marriage certificate, divorce certificates if either of you have been married previously, proof of your spouse's CU6 status, and proof of your legal entry into the U.S.
In about 3 to 5 months after filing, you should get a work permit. About 5 to 7 months after filing you should get a marriage interview. If all goes well, a few weeks later you should get a Residency status (green card).
Here is a link to information about the Cuban Adjustment:
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