  In this paper we study the behavior of the first Zassenhaus conjecture (ZC1)
under direct products as well as the General Bovdi Problem (Gen-BP) which turns
out to be a slightly weaker variant of (ZC1). Among others we prove that
(Gen-BP) holds for Sylow tower groups, so in particular for the class of
supersolvable groups. (ZC1) is established for a direct product of
Sylow-by-abelian groups provided the normal Sylow subgroups form together a
Hall subgroup. We also show (ZC1) for certain direct products with one of the
factors a Frobenius group. We extend the classical HeLP method to group rings
with coefficients from any ring of algebraic integers. This is used to study
(ZC1) for the direct product $G \times A$, where $A$ is a finite abelian group
and $G$ has order at most 95. For most of these groups we show that (ZC1) is
valid and for all of them that (Gen-BP) holds. Moreover, we also prove that
(Gen-BP) holds for the direct product of a Frobenius group with any finite
abelian group.
