Let $X$ be a smooth vector field on $\mathbb{T}^2$ without singular points and periodic orbits. Then by this theorem
Theorem:(Poincaré, Denjoy) Every non-singular $\mathcal{C}^2$ vector field on a compact surface that has no periodic orbits is topologically equivalent to a linear flow on a torus with the irrational slope,
$X$ is topologically equivalent to the irrational flow. It is trivial that if $Y$ is a irrational flow on $\mathbb{T}^2$ then there exists two $\mathcal{C}^1$ loops $$\alpha_1,\alpha_2 : \mathbb{S}^1\to \mathbb{T}^2$$ with $\alpha_1(0) = \alpha_2(0) = p$, such that $\{Y(\alpha_i(t)),\alpha_i'(t)\}$ is a basis of $T_{\alpha(t)} \mathbb{T}^2$, $\forall$ $t$ $\in$ $\mathbb{S}^1$, and $[\alpha_1 (t)], [\alpha_2 (t)]$ are generators of the group $\pi_1(\mathbb{T}^2,p)$.
My Doubt: Using that $X$ is topologically equivalent to $Y$, is it possible to guarantee that there exists two $\mathcal{C}^1$ loops $$\beta_1,\beta_2 : \mathbb{S}^1\to \mathbb{T}^2$$ with $\beta_1(0) = \beta_2(0) = p$, such that $\{X(\beta_i(t)),\beta_i'(t)\}$ is a basis of $T_{\alpha(t)} \mathbb{T}^2$, $\forall$ $t$ $\in$ $\mathbb{S}^1$, and $[\beta_1 (t)], [\beta_1 (t)]$ are generators of the group $\pi_1(\mathbb{T}^2,p)$?