Set of points dense in subset of four-dimensional space We may assume the following theorem:

Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$.

Consider the following situation:

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field $$w=x_1\frac\partial{\partial x_2}-x_2\frac\partial{\partial x_1}+\lambda\left(x_4\frac\partial{\partial x_3}-x_3\frac\partial{\partial x_4}\right)$$ ($\lambda\in\mathbb{R}$) is tangent to $X$ and hence defines by restriction a vector field $v$ on $X$. Suppose $\lambda$ is irrational. Prove that for every integral curve $\gamma(t)$ $( -\infty<t<\infty)$ of $v$, the set of points on this curve is a dense subset of $X$.

I computed the integral curve to be $$\gamma(t)=(a\cos t+b\sin t, a\sin t-b\cos t, c\cos(\lambda t)+d\sin(\lambda t), -c\sin(\lambda t)+d\cos(\lambda t))$$ for some constants $a,b,c,d$ where $a^2+b^2=c^2+d^2=1$.
Why must these points be dense in $X$?
 A: Assume first that $a = c = 1$ and $b = d = 0$. The corresponding integral curve is
$$
\gamma_0(t) = \bigl(\cos t, \sin t, \cos(\lambda t), -\sin(\lambda t)\bigr).
$$
Fix a real number $t_0$ arbitrarily, and consider the image of $t_0 + 2\pi \mathbf{Z}$, i.e., the set of points of the form
$$
\gamma_0(t_0 + 2\pi k) = \bigl(\cos t_0, \sin t_0, \cos(\lambda t_0 + 2\pi k\lambda), -\sin(\lambda t_0 + 2\pi k\lambda)\bigr)\quad\text{for some integer $k$}.
$$
This set consists of all images of the point $\bigl(\cos t_0, \sin t_0, \cos(\lambda t_0), -\sin(\lambda t_0)\bigr)$ under rotation (in the second circle factor) by multiples of $\lambda$ full turns; since $\lambda$ is irrational, this set is dense in the circle $(\cos t_0, \sin t_0, x_3, x_4)$ by the stated theorem.
Because the intersection of $\gamma_0(\mathbf{R})$ with an arbitrary circle $(\cos t_0, \sin t_0, x_3, x_4)$ is dense, $\gamma_0(\mathbf{R})$ is dense in the torus.
The general case may be handled by picking real numbers $\theta_1$ and $\theta_2$ such that $(a, -b) = (\cos\theta_1, \sin\theta_1)$ and $(c, -d) = (\cos\theta_2, \sin\theta_2)$, and noting that your integral curve is
$$
\gamma(t) = \bigl(\cos(t + \theta_1), \sin(t + \theta_1), \cos(\lambda t + \theta_2), -\sin(\lambda t + \theta_2)\bigr),
$$
i.e., is the image of $\gamma_0$ under a translation in the torus (viewing the torus as an additive group).
