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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. Wecharacterize all linear, quadratic, cubic Kolmogorov and homogeneous vectorfields on $S^1\times S^2$ and $S^2\times S^1$. We construct some firstintegrals of these vector fields and find which of the vector fields areHamiltonian. We give upper bounds for the number of the invariant meridian andparallel hyperplanes of these vector fields. In addition, we have shown thatthe upper bounds are sharp in many cases.

Quadratic, Homogeneous and Kolmogorov vector fields on $S^1\times S^2$ and $S^2 \times S^1$