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If you want to move out of the country and your ex who also has PR will not agree then you would need to apply for a specific issue order to move out for any period in excess of 1 month. Otherwise it can be considered as child abduction. As such if you can not get agreement then you would need to formally apply to the court for such an order.
To be perfectly honest it does not give any real greater powers or rights over the child. Really as you state to confirm that the child lives with you. One of the advantages/disadvantages of making an application for a child arrangement order ("live" with) is that it can cement in a court order what time the other parent spends with the child. Depending on circumstances this can be both helpful/unhelpful. A 'live' with order does not allow you to take the child out of the jurisdiction for periods in excess of 28 days. You would still need consent or a SPO. Please rate for me,