About generating set of the invariant subalgebra of free restricted Lie algebra

Suppose that $L=L(X)$ is the free Lie p-algebra of finite rank $k$ with free generating set $X=\{x_1,\dots,x_k\}$ on a field of positive characteristic. Let $G$ is nontrivial finite group of homogeneous automorphisms $L(X)$. Our main purpose to prove that $L^G$ subalgebra of invariants is is infinitely generated. We have more strongly result. Let $Y=\cup_{n=1}^\infty Y_n$ be homogeneous free generating set for the algebra of invariants $L^G$, elements $Y_n$ are of degree $n$ relatively $X$, $n\ge1$. Consider the corresponding generating function $\mathscr H(Y,t)=\sum_{n=1}^\infty|Y_n|t^n$. In our case of free Lie restricted algebras, we prove, that series $\mathscr H(Y,t)$ has a radius of convergence $1/k$ and describe its growth at $t\to1/k-0$. As a result we obtain that the sequence $|Y_n|$, $n\ge1$, has exponential growth.