  In this paper, we classify all cubic vector fields in $\mathbb{R}^3$ such
that 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 $\mathbb{T}^2$ and prove
that they exhibit a rational first integral. We explicitly find all cubic
vector fields on $\mathbb{T}^2$ which achieve the sharp upper bounds for the
number of invariant meridians and parallels. We present necessary and
sufficient conditions when invariant meridians and parallels of cubic vector
fields on $\mathbb{T}^2$ are periodic orbits or limit cycles. We study `Pseudo
Type-$n$' vector fields on $\mathbb{T}^2$ and show that any such vector field
is completely integrable. We discuss invariant meridians and parallels of
Pseudo Type-$n$ vector fields also. We prove that Lie bracket of any two
quadratic vector field on $\mathbb{T}^2$ is completely integrable. Moreover,
Lie bracket of two Pseudo Type-2 vector fields on $\mathbb{T}^2$ can have at
most two invariant meridians.
