Hello, my name is ***** ***** I will be the immigration attorney helping you today.
As a green card holder (permanent resident) of the US, you are supposed to be living in the US. In order to know whether you are living in the US, it is required that you spend more time inside the US than outside the US. Your plan of spending a month in the US and then 2 months outside, may work for some time, but eventually the Customs officers at the airport are going to begin questioning whether you are actually living in the US and may recommend that you stop spending so much time outside the US. It will be more of a risk the more you do it and it starts to become a pattern.
One temporary option that you have to maintain your green card will be to apply for a "reentry permit" once you have your green card. A reentry permit allows green card holders to travel in and out of the US without having to return to the US for up to 2 years without losing your green card. If you got it then you would not have to worry about coming in and out and risking your green card. However, this is not a permanent solution. You can get a few re-entry permit for two years each but then after that you will only get it for one year and there is never a guarantee that it will be approved the next time. To apply for that you must use application I-131 on uscis.gov/forms. You must be in the US when you apply and then it will take about 2-3 months to arrive. You don't have to be in the US to wait for it but you should make sure that someone reliable mails it to you.
Good luck to you during this process and I hope this answer has helped you!
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