For a Lie algebra \(L\), \(U(L)\) (respectively \(u(L)\)) denotes the (restricted) universal enveloping algebra of \(L\). The main results of the article are the following two theorems. Theorem 2.3. Let \(G\) be a group, \(L\) be a Lie algebra over a field \(K\) of characteristic \(p>0\) and \(G\) act on \(L\) by automorphisms. Then the smash product \(U(L)\sharp K[G]\) satisfies a nontrivial polynomial identity if and only if the following holds: 1) there exists an abelian \(G\)-invariant ideal \(H\subset L\) of finite codimension and all derivatives \(\text{ ad} x\), \(x\in L\), are algebraic of bounded degree; 2) there exists a normal subgroup \(A \subset G\) of finite index with the commutator subgroup \(A'\) being a finite abelian \(p\)-group; 3) \(A\) acts trivially on \(L\). Theorem 3.1. Suppose that a group \(G\) acts by automorphisms on a Lie \(p\)-algebra \(L\). Then the smash product \(u(L)\sharp K[G]\) is a PI-algebra if and only if the following holds: 1) there exist \(G\)-invariant restricted subalgebras \(Q\subset H\subset L\) with \(\text{ dim} L/H<\infty\), \([H,H]\subset Q\), and \(Q\) is abelian with a nilpotent \(p\)-mapping; 2) there exists a subgroup \(A\subset G\) with \(|G:A|<\infty\) and the commutator subgroup \(A'\) is a finite abelian \(p\)-group; 3) \(A\) acts trivially on \(H/Q\).