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