  In general, for a bipartite quantum system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and an integer $k$ such that $4\leq k\le
d$,there are few necessary and sufficient conditions for local discrimination
of sets of $k$ generalized Bell states (GBSs) and it is difficult to locally
distinguish $k$-GBS sets.The purpose of this paper is to completely solve the
problem of local discrimination of GBS sets in some bipartite quantum
systems.Firstly three practical and effective sufficient conditions are
given,Fan$^{,}$s and Wang et al.$^{,}$s results [Phys Rev Lett 92, 177905
(2004); Phys Rev A 99, 022307 (2019)] can be deduced as special cases of these
conditions.Secondly in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, a necessary and
sufficient condition for local discrimination of GBS sets is provided, and a
list of all locally indistinguishable 4-GBS sets is provided,and then the
problem of local discrimination of GBS sets is completely solved.In
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, a concise necessary and sufficient
condition for one-way local discrimination of GBS sets is obtained,which gives
an affirmative answer to the case $d=5$ of the problem proposed by Wang et al.
