  In ~\cite{Iw2} Iwase has constructed two 16-dimensional manifolds $M_2$ and
$M_3$ with LS-category 3 which are counter-examples to Ganea's conjecture:
${\rm cat_{LS}} (M\times S^n)={\rm cat_{LS}} M+1$. We show that the manifold
$M_3$ is a counter-example to the logarithmic law for the LS-category of the
square of a manifold: ${\rm cat_{LS}}(M\times M)=2{\rm cat_{LS}} M$. Also, we
construct a map of degree one $$f:N\to M_2\times M_3$$ which reduces Rudyak's
conjecture to the question whether ${\rm cat_{LS}}(M_2\times M_3)\ge 5$. We
show that ${\rm cat_{LS}}(M_2\times M_3)\ge 4$.
