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Your question seems fairly short and simple, so forgive me if my answer seems simple as well and you can ask any specific follow up questions you might have, as you likely know.
Yes, the NY courts whenever possible prefer shared custody, which is joint legal/physical custody over a child where parents share physical custody time with the child. In order for the courts to do this, they must be satisfied that the parents will be able to work together towards the best interests of the child. If the courts do not find that the best interests of the child will be served by shared custody, they would name one parent primary physical custodian and then have joint legal custody where both parents are responsible for legal determinations involving the child.
At two years old, sometimes you face prejudice of many judges who believe that young children belong with the mother, but the law itself does not prevent you from having joint physical/legal custody over the child at all and as long as you can prove you and the other parent can work together concerning the child, the court would grant such custody.
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