  We first prove, for pairs consisting of a simply connected complex reductive
group together with a connected subgroup, the equivalence between two different
notions of Gelfand pairs. This partially answers a question posed by Gross, and
allows us to use a criterion due to Aizenbud and Gourevitch, and based on
Gelfand-Kazhdan's theorem, to study the Gelfand property for complex symmetric
pairs. This criterion relies on the regularity of the pair and its descendants.
We introduce the concept of a pleasant pair, as a means to prove regularity,
and study, by recalling the classification theorem, the pleasantness of all
complex symmetric pairs. On the other hand, we prove a method to compute all
the descendants of a complex symmetric pair by using the extended Satake
diagram, which we apply to all pairs. Finally, as an application, we prove that
eight out of the twelve exceptional complex symmetric pairs, together with the
infinite family $(\textrm{Spin}_{4q+2}, \textrm{Spin}_{4q+1})$, satisfy the
Gelfand property, and state, in terms of the regularity of certain symmetric
pairs, a sufficient condition for a conjecture by van Dijk and a reduction of a
conjecture by Aizenbud and Gourevitch.
