The algorithms that recognize Milnor laws and properties of these laws

We consider several equivalent definitions of the so-called Milnor laws (or Milnor identities) that is the laws which are not satisfied in (formula presented) varieties. The purpose of this article is to provide algorithms that allow us to check whether a given identity w(x, y) has one of the following properties: • w(x, y) is a Milnor law, • every nilpotent group satisfying w(x, y) is abelian, • every finitely generated metabelian group satisfying w(x, y) is finite-by-abelian.