Dear @Bright_love,
Unless you have other residency opportunities (e.g. being a skilled worker), your residency rights are dependent on the residency situation of your children. If they continue to enjoy EU freedom of movement rights, then you have the right to a residency permit according to this rule of EU law: Article 20 of the Treaty on the Functioning of the European Union. This rule gives non-EU citizen parents the right to reside in Germany based on the fact that they care for the children and/or have legal rights regarding access to and custody of the children. It is not required to be working or have a certain amount of money to have this right.
However, as their mother have moved away, it is unlikely they continue to enjoy freedom of movement rights. For children this is most commonly the case when an EU citizen parent is working and as result the children enjoy the freedom of movement rights as family members of a person who is working.
There is a special rule regarding children who are attending school: If their EU parent at one time worked in Germany, then the children have a special right to stay in Germany until their school education has ended. It does matter for how long the EU citizen parent worked or how the work came to an end. If this is the situation for at least one of your children, then they have this special residency right. It is according to article 10 of the EU regulation No 492/2011. However, I am only certain of this right applying if the EU citizen remains in Germany and I also do not know if this type of EU residency is sufficient for you to receive the Article 20 residency as outlined above.
Based on the information you have provided, in my view the special rule Article 20 is the most likely possibility for you to receive a residence permit.
The relevant German ministry has published information on this special Article 20 rule and it says that for cases where the relationship is only between the non-EU parent and child, then that parent can (not must) be refused a permit in Germany. This can only happen if the non-EU parent has residency rights in the country of which the child is a national. That means that if you are not entitled to residency in the Czech Republic, then Germany will have to provide you with a permit.
In my opinion, the best course of action would be to make a written application for this special type of residency permit with your local immigration office (Ausländerbehörde). You should include documents regarding your custody rights and, if applicable, proof of any child attending school (you can get a paper called a ‘Schulbescheinigung’ from the school). Once you have made an application in writing, the Ausländerbehörde are obliged to examine whether you have the residency rights as outlined above and provide you with a written response.
As my colleague wrote, there are specialised services throughout Germany who can help you with this. It is important that anyone helping you is familiar with immigration law around family members of EU citizens. There may be other people, however, who see your case differently and have other ideas to me.
I hope the above makes sense. Unfortunately, it is a very complicated part of the immigration law. If there’s anything you do not understand, let us know.
Best,
Éanna