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In this paper, we classify all cubic vector fields in $\mathbb{R}^3$ suchthat their flow makes the torus $$\mathbb{T}^2=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2-a^2)^2+z^2-1=0\}~\mbox{with}~a\in (1,\infty)$$ invariant.We also classify cubic Kolmogorov vector fields on $\mathbb{T}^2$. Weinvestigate invariant meridians and parallels of cubic vector fields andpresent some necessary and sufficient conditions when they will be periodicorbits or limit cycles. We study `Pseudo Type-$n$' vector fields and cubicvector fields generated by Lie bracket of two arbitrary quadratic vector fieldson $\mathbb{T}^2$. We discuss invariant meridians and parallels of PseudoType-$n$ vector fields also.

Certain classes of vector fields on the torus and some of their periodic orbits