The sociopath only wants to WIN

If you are a woman and are dealing with a sociopath ex. If you are lucky, your ex will find a new lover and become uninterested in seeing his children. Sociopath/narcissistic men usually are selfish and children are more of an inconvenience to them.
When a sociopathic male tries to seek custody of children , they do so because they want to either
1. Hurt the other parent.
2. If they make good money and are going to have to pay a large amount of child support.
If the sociopathic male is wealthy and has plenty of money to take you to court and fight for custody of the children - HE WILL DO SO. And of course if he is able to discredit the mother with lies and slander, and gets somewhat custody of the children, you can bet that he will not be the one who is taking care of the kids. He will have a relative take care of them or find a women who will take care of them.
He only wanted to take the children from the mother in order TO WIN.

That is what it is all about. WINNING. Male or Female. The sociopath wants to win. They want to turn you into dust. They will usually boast about how they have custody or how they are the "domiciliary" parent. They portray themselves as Mom or Dad of the year. Always talking about how they do this and that for their kids, and letting everyone know what a piece of shit the other parent is.

When dealing with a sociopath, in my experience NO CONTACT is the only option. Of course if you share children with the sociopath this becomes a bit more complicated. 

If you are a man, and you ex is a sociopath, this situation is difficult and frustrating. 
When children are younger, in order to see the children, communication will have to be made. Every interaction with the sociopath is going to be chaotic, frustrating, and stressful. 
But I can tell you that it does not have to last like this forever. 
There  is a light at the end of what seems like a very long, hopeless tunnel. 
Your children will get older. And soon you will be able to communicate directly with them. The sociopath will not like this at all. They will tell  you that you are an irresponsible parent for "using" the children to communicate with. They will give you lectures on how you should coparent with them. They will make desperate attempts to speak with you. They will use the kids to try to talk with you. They will make up lies about the children.  
Example: they will tell you that a child is having problems in school due to your lack of co-parenting, or that the child is going thru hormonal changes and they want you to talk with them about boys/girls, birds and bees, etc... Blah blah blah blah....

A close friend of mine dealt with a sociopathic ex for years, but started communicating with kids by cell phone he purchased for kids. The ex was losing control, but couldn't take kids phone because all other kids that age had phones, and she didn't want to look like the bad guy by taking it. But she was angry that she was unable to speak to him and manipulate and waste his time by using the kids. He was able to make plans to get the kids on his weekends, without having to go through the crap that came with dealing with the ex. She loved to waste his time by taking the children else where when he was suppose to pick up. 
 But if your child is able to communicate with you via cell phone/text, you will be able to confirm with them if what the other parent is telling you is true or false . 
Remember they only want direct communication with you, because without it they cannot manipulate you and use the children to waste your time and drain your energy and wallet.