In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977
"The Products of Manifolds with the f.p.p. Need Not have the f.p.p"
who gave an example of two fixed point manifolds $M,N$ such that $M\times N$ is not a fixed point space. One can consider the fixed point property as a kind of a problem in discrete dynamical system. So in this question we would like to consider a continuous dynamical version of this result by S.Husseini. May be a flow version of the concept of fixed points is the concept of closed orbits, compact leaves of the 1-dimensional foliation associated to the (non vanishing) vector field which generate our given flow.
The above is a motivation to ask the following question:
What is an example of two compact manifolds $M,N$ whose product $M\times N$ admits a foliation without compact leaf but every foliation of $M$, and every foliation of $N$ has at least one compact leaf.
Note: In this question we do not specify the dimension of the foliations under consideration but we require that the foliation has non trivial dimension. Neither 0 dimensional nor the full dimensional foliation.
So in this question the obvious answer $S^1 \times S^1$, as the ambient space of the kronecker foliation, is excluded. Since $S^1$ foliates itself by a single leaf, a trivial foliation.
