Invariant Parallels, Invariant Meridians and Limit Cycles of Polynomial Vector Fields on Some 2-Dimensional Algebraic Tori in \mathbb R ^3

We consider the polynomial vector fields of arbitrary degree in $\mathbb R ^3$ R 3 having the 2-dimensional algebraic torus $$\begin{aligned} \mathbb T ^2(l,m,n)=\{(x,y,z)\in \mathbb R ^3 : (x^{2l}+y^{2m}-r^2)^2+z^{2n}-1=0\}, \end{aligned}$$ T 2 ( l , m , n ) = { ( x , y , z ) ∈ R 3 : ( x 2 l + y 2 m - r 2 ) 2 + z 2 n - 1 = 0 } , where $l,m$ l , m , and $n$ n positive integers, and $r\in (1,\infty )$ r ∈ ( 1 , ∞ ) , invariant by their flow. We study the possible configurations of invariant meridians and parallels that these vector fields can exhibit on $\mathbb T ^2(l,m,n)$ T 2 ( l , m , n ) . Furthermore, we analyze when these invariant meridians or parallels are limit cycles.