  In this paper, we consider the following two algebraic hypersurfaces
$$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2
+ x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in
\mathbb{R}^4:(x_1^2+x_2^2+x_3^2-b^2)^2+x_4^2-1=0;~ b>1\}$$ embedded in
$\mathbb{R}^4$. We study polynomial vector fields in $\mathbb{R}^4$ separately,
having $S^1\times S^2$ and $S^2\times S^1$ invariant by their flows. We
characterize all linear, quadratic, cubic Kolmogorov and homogeneous vector
fields on $S^1\times S^2$ and $S^2\times S^1$. We construct some first
integrals of these vector fields and find which of the vector fields are
Hamiltonian. We give upper bounds for the number of the invariant meridian and
parallel hyperplanes of these vector fields. In addition, we have shown that
the upper bounds are sharp in many cases.
