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Consider the critical Galton-Watson branching system with infinite varianceof the offspring law. We provide an alternative arguments against whatSlack~{\cite{Slack68}} did when it seeked for a local expression in theneighborhood of point $1$ of the generating function for invariant measures ofthe branching system. So, we obtain the global expression for all $s\in[0,1)$of this generating function. A fundamentally improved version of thedifferential analogue of the Basic Lemma of the theory of critical branchingsystems is established. This assertion plays a key role in the formulation ofthe local limit theorem with explicit terms in the asymptotic expansion oflocal probabilities. We also determine the decay rate of the remainder term inthis expansion.

Further remarks on the explicit generating function expression of the invariant measure of critical Galton-Watson Branching Systems