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Alienation refers to parental alienation. This means if one parent poisons the child's relationship with the other non custodial parent. There's actually a diagnosis called Parental Alienation Syndrome which is disputed as to whether or not it even exists by the medical profession. One of the first rules in dealing with parental alienation is that it does not apply where there is a history of physical abuse. So if there is a history of physical abuse by Father that needs to be raised before the court.
Filing a motion for reconsideration and appeal is a complicated process and technical process. This website is designed to answer basic legal questions. It is not a good idea to try to utilize an attorney on this website to file something as complicated as a motion for reconsideration or an appeal. Both a motion for reconsideration and appeal need to be filed simultaneously as well as the ordering of the transcript from the preceding. From your basic description of the situation that does sound like mother needs to take these two steps, but if she's going to do it hire an attorney familiar with this particular judge and county because if your appeal is granted the matter will be remanded back down to the trial judge to take additional evidence and formulate a different order. You also want to ensure that the attorney you hire is it family law attorney experience in dealing with appeals. Hence, she will ultimately still be dealing with the same judge. In addition to doing a motion for reconsideration and an appeal I would recommend requesting a guardian ad litem be appointed on behalf of the children if it has not already been done. A guardian ad litem is an attorney who represents the best interest of the children not mother or father. Guardian ad litem can provide much more detail to the court as they can conduct home inspections, and speak to parents, teachers, doctors, friends, neighbors, counselors, etc. The guardian ad litem is an excellent way to refute the allegation of parental alienation.
I hope that this information and suggestions are helpful and I wish you the best of luck.