I believe the answer to both of your questions is ''yes".
Any real-analytic vector field on $\mathbb{S}^2$ that has a
center singularity can be lifted to a one-parameter family of
non-vanishing vector fields on $\mathbb{S}^3$ via the Hopf
fibration map so that any vector field from the family has an
invariant foliation of tori on which the dynamics is real-analytically conjugate to a linear dynamics on each torus
from the foliation and thus, there is an infinite (like the
irrational numbers) family of tori with the Kroneker's foliation
dynamics of varying angles. (It's kind of like a
Liouville-Arnold's theorem).
The three-sphere $\mathbb{S}^3$ is a Lie group and can be
seen for example as the unit quaternions (probably the most
convenient description) as well as the matrix group $\text{SU}(2)$, which is
more or less equivalent to the quaternions. Let $\pi :
\mathbb{S}^3 \to \mathbb{S}^2$ be the Hopf fibration map, which is
a principle bundle map with an $\mathbb{S}^1$ action on
$\mathbb{S}^3$. Let's say $\mathbb{S}^1$ acts on the right, which
means the Hopf map can be seen for instance as $\phi \in
\mathbb{S}^3 \, \mapsto \, \pi(\phi) = \phi \, i \,\phi^{-1} \in
\mathbb{S}^2$ for $i \in \mathbb{S}^2$ in quaternionic
notation, which is isomorphic to the classical representation $
\phi = w + j \, z \in \mathbb{S}^3 \, \mapsto \, z/w \, \in
\mathbb{C} \cup \{\infty\} \cong \mathbb{S}^2$, also known as
$[z:w] \in \mathbb{CP}^1\cong \mathbb{S}^2,$ via stereographic
projection. In this picture $\mathbb{S}^1$ is the stabilizer of
$i$ and thus its elements are given as the usual $e^{i\theta}$ and
their action is simply quaternionic multiplication on the right
$\phi \mapsto \phi \, e^{-i\theta}$. The Lie algebra of
$\mathbb{S}^3$ is the tangent space $T_1\mathbb{S}^3$ at the
identity element $1 \in \mathbb{S}^3$ and is isomorphic to the
imaginary quaternions $\text{Im}(\mathbb{H})$. Now use the
standard dot product on $T_1\mathbb{S}^3$, which is the same as
the Killing form of the algebra, to split it orthogonally into
$T_1\mathbb{S}^3 = \mathbb{R}i \oplus E_1$, where $E_1 =
\mathbb{R}j \oplus \mathbb{R}k$. Recall that the left translation
on the group is $L_{\phi}(\psi) = \phi\psi$ and its tangent map is
$(DL_{\phi})_{\psi} : T_{\psi}\mathbb{S}^3 \to
T_{\phi\psi}\mathbb{S}^3$ . Then, define the left-invariant vector
field $V({\phi}) = (DL_{\phi})_1 i$ which generates the
$\mathbb{S}^1$ action on the three-sphere and is thus tangent to
the circular fiber of the Hopf bundle (right action is left
invariant because right and left commute). Therefore $\ker
(D_{\phi}\pi) = \mathbb{R} \, V(\phi)$ is the one-dimensional
kernel of the tangent map to the Hopf map. Similarly, define the
plane field $E_{\phi} = (DL_{\phi})_1 E_1$ (i.e. a two dimensional
distribution or if you prefer a two dimensional subbundle of the
tangent bundle of $\mathbb{S}^3$) to obtain the tangent bundle
splitting $T_{\phi}\mathbb{S}^3 = \mathbb{R} \, V(\phi) \oplus
E_{\phi} = \ker (D_{\phi}\pi) \oplus E_{\phi}$. Hence, the
restriction $(D_{\phi}\pi)|_{E_{\phi}} \, : \, E_{\phi} \, \to \,
T_{\pi(\phi)}\mathbb{S}^2$ is onto and has no kernel so it is an
isomorphism at each point $\phi \in \mathbb{S}^3$. Now, let your
real-analytic vector field on the two-sphere $\mathbb{S}^2$ be
$X(x)$ and for each $\phi \in \mathbb{S}^3$ define the tangent
vector
$$Y(\phi) = \Big( (D_{\phi}\pi)|_{E_{\phi}}\Big)^{-1}
X(\pi(\phi)).$$ This construction defines a real-analytic lift of
$X$, that is $Y(\phi)$ is a real-analytic vector field on the
three sphere such that $(D_{\phi}\pi)\, Y(\phi) = X(\pi(\phi))$,
which means $Y$ projects down on $X$ via $\pi$. Now, by
construction, the distribution $E_{\phi}$ is left-invariant but it
is not integrable. It actually defines a contact structure on the
three-sphere. The bracket $[V, Y](\phi)$ belongs to $E_{\phi}$ for
any $\phi \in \mathbb{S}^3$. Indeed, if $J(\phi)=(DL_{\phi})_1j$
and $K(\phi)=(DL_{\phi})_1k$, then the left-invariant vector
fields $J$ and $K$ define a global basis for the distribution $E$.
Moreover, $[V, J]=K$ and $[V, K]=-J$. Therefore, $Y(\phi) =
\alpha(\phi)J(\phi)+\beta(\phi)K(\phi)$ and thus $$[V, Y] =
(\mathcal{L}_V\alpha)J + \alpha[V, J] + (\mathcal{L}_V\beta)K +
\beta[V, K] = (\mathcal{L}_V\alpha - \beta)J + (\mathcal{L}_V\beta
+ \alpha)K.$$ Here $\mathcal{L}_V\alpha$ is the Lie derivative of
$\alpha$ in direction of $V$, etc. By construction, $Y$ projects
on $X$ and therefore the orbits of $Y$ project onto the orbits of
$X$ which are either points, embedded closed circles or immersed
real lines. The lifts of the fixed points of $X$ are Hopf circles
made of fixed point singularities of $Y$. Let us now focus on the
other two types of non-trivial orbits of $X$: embedded circles and
immersed real lines (embedded is a special case of immersed). The
lift of each of these orbits via $\pi$ on $\mathbb{S}^3$ is either
(i) a torus embedded in the three sphere, in the case of a closed
orbits, or (ii) immersed cylinder, in the case of an immersed
line. Thus we obtain a two-dimensional real analytic foliation on
$\mathbb{S}^3$ (with circular singularities), invariant under the
$\mathbb{S}^1$ action, i.e. each of its leaves foliates into Hopf
fibers over an orbit of $X$. Now, in addition to that, we know
that $X$ has center dynamics (possibly local) and thus has an
analytic family of closed orbits circling around the center. As
already discussed, the lift of each of these orbits via $\pi$ on
$\mathbb{S}^3$ is a torus embedded in the three sphere. Thus, we
obtain a real analytic foliation (possibly local) of embedded
(nested) tori, invariant under the $\mathbb{S}^1$ action, so that
each of them is foliated into Hopf fibers over a periodic orbit of
$X$ around the center. Denote by $\text{T}_{\phi}$ the unique
torus passing thorough a point $\phi \in \mathbb{S}^3$, whenever
defined. Also, denote by $\text{S}_{\phi}$ the unique Hopf fiber
passing through $\phi \in \mathbb{S}^1$. By construction, both $Y$
and $V$ are tangent to the non-singular leaves of the toric
foliation $\text{T}_{\phi}$ (and actually of the total foliation
of tori and cylinders obtained by lifting of the orbits of $X$),
spanning their tangent planes and forming a plane field (a
distribution with singularities) tangent to the foliation. Hence,
this distribution is integrable by Frobenius, so $[V, Y](\phi) \,
\in \, \mathbb{R} \, Y(\phi) \oplus \mathbb{R} \, V(\phi)$.
Consequently, $[V, Y](\phi)$ belongs to the intersection of the
transverse (but not complimentary) tangent planes $E_{\phi}$ and
$\mathbb{R} \, Y(\phi) \oplus \mathbb{R} \, V(\phi)$. Now, since
$\big(\mathbb{R} \, Y(\phi) \oplus \mathbb{R} \, V(\phi) \big)
\cap E_{\phi} = \mathbb{R} \, Y(\phi)$, one can immediately
conclude that $[V, Y](\phi) = \lambda(\phi)Y(\phi)$ for some
real-analytic function $\lambda(\phi)$ defined on $\mathbb{S}^3$
minus the singular Hopf fibers. Finally, apply the map $D\pi$ to
the bracket $[V, Y]$ to obtain $$\lambda \, X = \lambda (D\pi) Y =
(D\pi)(\lambda Y) = D\pi [V, Y] = [(D\pi)V, (D\pi)Y] = [0, X] =
0.$$ This is possible if and only if $\lambda \equiv 0$.
Therefore, $[V, Y] = 0$.
Remark: There could be a more direct argument proving that
$[V, Y] = 0$. The Killing form, which is the unique
$\text{Ad}$-invariant dot product on the Lie algebra
$T_1\mathbb{S}^3$ defines the round metric on $\mathbb{S}^3$ given
by left-translations of the Killing form. The
$\text{Ad}$-invariance means that the metric is also
right-invariant, i.e. it is bi-invariant. As $E_\phi$ is defined
as the orthogonal complement of $V(\phi)$, then the splitting,
which is defined to be left-invariant, is also right-invariant
when acted upon with elements of $\mathbb{S}^1$. Therefore, the
lift of $X$ on $E$ via $\pi$ is $\mathbb{S}^1$ invariant, which
means that the generating vector field $V$ of the action commutes
with $X$.
Recall that the Riemannian metric on the three-sphere is
$$(u,v)_{\phi} = ((DL_{\phi^{-1}})_{\phi} \, u, (DL_{\phi^{-1}})_{\phi} \, v)_1$$
for the Killing form/dot product $( \cdot , \cdot )_1$ on the Lie
algebra (which is actually the standard dot product in
$\mathbb{R}^3$). The metric is bi-invariant (both left and right).
The Lie bracket on the Lie algebra $T_1\mathbb{S}^3$ is denoted by
$[\cdot, \cdot]_1$ and is actually the standard cross-product in
$\mathbb{R}^3$. Although it is the left projection of the vector
field Lie bracket of left-invariant vector fields, it is not equal
to the projection of the vector field Lie bracket of non-invariant
vector fields. That's why the superscript $1$, in order to
distinguish it from the vector field bracket.
Define the vector field $$\tilde{W}(\phi) = (DL_{\phi})_1
\Big(\text{ad}(i) \circ (DL_{\phi^{-1}})_{\phi} \, Y(\phi) \Big) =
(DL_{\phi})_1 \, [i, (DL_{\phi^{-1}})_{\phi} \, Y(\phi)]_1. $$ By
construction of $Y$, the new field $\tilde{W}$ is nonzero,
whenever $Y$ is non-zero and is orthogonal to both $Y$ and $V$.
Next, normalize it to the unit vector field $W(\phi) =
\tilde{W}(\phi) / \|\tilde{W}(\phi)\|$ defined on the three-sphere
minus some Hopf fiber singularities. It's an analytic unit vector
field orthogonally transverse to the toric foliation
$\text{T}_{\phi}$. We assume it is pointing inwards towards the
singularity of the toric foliation (if not, just take $-W$). Let
$\phi_0(s) \, : \, s \in (L,\infty)$ be one integral curve of $W$,
i.e. $\dot{\phi}_0(s) = W(\phi_0(s))$, with the property that its
projection $\pi(\phi_0(s))$ on $\mathbb{S}^2$ converges towards
the central singularity of $X$ when $s \to \infty$. By
construction, $\phi_0(s)$ is transverse to each $\text{T}_{\phi}$
and intersects it at exactly one point. Hence, we can rewrite the
analytic foliation $\text{T}_{\phi}$ as a real-analytic one
parameter family of tori $\text{T}_{s} = \text{T}_{\phi_0(s)}$.
Denote by $\Psi^{t_1}(\phi)$ the phase flow of the vector field
$Y$, i.e. $$\frac{d}{d t_1} \Psi^{t_1}(\phi) =
Y\big(\Psi^{t_1}(\phi)\big)\, , \,\,\, \Psi^0(\phi) = \phi.$$
Observe that $\phi e^{i t_2}$ is the phase flow of $V$ by
construction. Furthermore, define the real analytic map $F \, : \,
\mathbb{R}^2 \times (L,\infty) \, \to \, \mathbb{S}^3$ by
$$F \, : \, (t,s) = (t_1,t_2,s) \, \mapsto \,
\Psi^{t_1}\big(\phi_0(s)\, e^{it_2}\big) =
\Psi^{t_1}\big(\phi_0(s)\big) \, e^{it_2},$$ where the last
equality holds due to the commutation between $X$ and $V$. A
straightforward computation shows that
\begin{align*}
(D_{t,s}F)\frac{\partial}{\partial t_1} &= Y(F(t,s))\, , \\
(D_{t,s}F)\frac{\partial}{\partial t_2} &= V(F(t,s))\, , \\
(D_{t,s}F)\frac{\partial}{\partial s} &=
(DR_{e^{-it_2}}) \circ (D\Psi^{t_1})\, W(\phi_0(s)) =
(D\Psi^{t_1}) \circ (DR_{e^{-it_2}}) \, W(\phi_0(s)),
\end{align*}
all three of them being linearly independent. That mean that $F$ has a
maximal rank and is therefore a real-analytic local
diffeomorphism. It is onto its image and as such is actually a
universal covering map onto $F\big(\mathbb{R}^2 \times (L,
\infty)\big)$. It also shows that $F$ maps the straight constant
vector field $\partial/\partial t_1$ to $Y$ and $\partial/\partial
t_2$ to $Y$.
One can see that $s$ can be expressed as a function of $\phi$ as
follows. For $\phi \in \mathbb{S}^3$, take the unique torus
$\text{T}_{\phi}$ from the toric foliation (as long as $\phi$ is
in the domain of the toric foliation). By construction, it
intersects the curve $\phi_0(s)$ at a unique point, determined by
a unique $s$. Thus, set $s=s(\phi)$. The function $s(\phi)$ is
real analytic because it is the $s-$coordinate function of the
inverse map $F^{-1}$ (existing locally). It is also a first
integral of $Y$ determining the toric foliation as its level
surfaces.
Let $\delta_s(t_1)$ be an integral curve of $Y$ passing
through the point $\phi_0(s)$, i.e. $\dot{\delta}_s(t_1) =
Y\big(\delta_s(t_1)\big)$ and $\delta_s(0) = \phi_0(s)$. It lies
on $\text{T}_s$. Denote by $\delta_s$ the part of the orbit
$\delta_s(t_1)$ which starts from $\phi_0(s)$ and goes around the
torus until it comes back and intersect the Hopf fiber
$\text{S}_{\phi_0(s)}$ again for the first time (not counting the
initial staring point $\phi_0(s))$. In other words, this is a
Poincar\'e map construction. Denote the first return point by
$\phi_1(s)$. Now let $\gamma_s$ be the simple arc on the Hopf
fiber $\text{S}_{\phi_0(s)}$ starting from $\phi_1(s)$ and ending
at $\phi_0(s)$ following the orientation provided by the vector
field $V$ which is tangent to the fiber. Consequently, we have a
(piecewise smooth) simple closed loop $\delta_s \cup \gamma_s$ on
the torus $\text{T}_s$ from $\phi_0(s)$ to itself. Since $F$ is a
universal covering map, by the path lifting property of covering
spaces, lift $\delta_s \cup \gamma_s$ on the universal cover
$\mathbb{R} \times (L,\infty)$, which actually ends up being a
curve on $\mathbb{R} \times \{s\}$, starting from $(0,0,s)$ to the
end point $(a_1(s),a_2(s), s) \neq (0,0,s)$ depending
real-analytically on $s$. Define the tangent vector
$a(s)=(a_1(s),a_2(s),0)$ at $(0,0,s)$. If we denote by $e_2$ the
vector $2\pi \partial/\partial t_2$ or in coordinates $(0,2\pi,0)$
at $(0,0,s)$, then $F \, : \, \left(\mathbb{R}^2 / \big(\mathbb{Z}
\, a(s) \oplus \mathbb{Z} \, e_2 \big) \right) \times \{s\} \, \to
\, \text{T}_s$ is a real analytic diffeomorphism between tori.
Then, let us change the coordinates by the linear transformation
\begin{align*}
t_1 &= a_1(s) \sigma_1 \\
t_2 &= a_2(s) \sigma_1 + 2\pi \sigma_2.
\end{align*}
Then one goes from $(t,s) = (t_1,t_2,s)$ coordinates to $(\sigma, s) = (\sigma_1, \sigma_2, s)$ coordinates. Denote
by $A(s)$ the inverse linear map of that transformation, i.e. $A(s)$ is given by
\begin{align*}
\sigma_1 &= \frac{1}{a_1(s)} t_1 \\
\sigma_2 &= \frac{a_2(s)}{2\pi a_1(s)} t_1 + \frac{1}{2\pi} t_2.
\end{align*}
For simplicity, let $\omega_1(s) = 1/a_1(s)$ and $\omega_2(s) = a_2(s)/(2\pi
a_1(s))$. Then it is immediate to see that under the
transformation $A(s)$ we can rewrite the vector fields as follows:
$$\frac{\partial}{\partial t_1} = \omega_1(s) \frac{\partial}{\partial
\sigma_1} + \omega_2(s) \frac{\partial}{\partial \sigma_2}\, \, \,\,
\text{ and } \,\,\,\, \frac{\partial}{\partial t_2} = \frac{1}{2\pi}
\frac{\partial}{\partial \sigma_2}.$$ As $a_1(s)$ and $a_2(s)$
depend real analytically on $s$, so do $\omega_1(s)$ and
$\omega_2(s)$ (the denominator $a_1(s)$ is always positive so
never zero, because it is actually the time of first return of the Poincar\'e map). Now, define the new real-analytic map (a local
real-analytic diffeomorphism and a universal covering map onto its
image)
\begin{align*}
\Phi \, &: \, \mathbb{R}^2 \times (L,\infty) \, \to \, \mathbb{S}^3 \\
\Phi \, &: \, (\sigma, s) \, \to \, F\big( A(s)\sigma, s \big).
\end{align*}
Finally, let us define the one parameter family of vector fields
$\tilde{X}_{\varepsilon}(\phi) = Y(\phi) + \varepsilon \,
V(\phi)$, then by construction, $(D_{\phi}\pi)\big(Y(\phi) +
\varepsilon \, V(\phi)\big) = X(\pi(\phi))$ for any $\phi \in
\mathbb{S}^2$. The new vector field $\tilde{X}_{\varepsilon}$ is
non-vanishing on the three-sphere for $\varepsilon \neq 0$ and is
tangent to the toric foliation $\text{T}_{\phi}$. Also, written in
$(\sigma, s)$ coordinates, it becomes
$$(D_{\phi}\Phi^{-1}) (\tilde{X}_{\varepsilon}(\phi)) =
(D_{\phi}\Phi^{-1}) \big(Y(\phi) + \varepsilon \, V(\phi)\big) =
\omega_1(s) \frac{\partial}{\partial \sigma_1} + \left(\omega_2(s)
+ \frac{\varepsilon}{2\pi}\right) \frac{\partial}{\partial
\sigma_2}.$$ If we set $\omega_2(s,\varepsilon) = \omega_2(s) +
\varepsilon/{2\pi}$, its orbits satisfy the equations
\begin{align*}
\dot{\sigma}_1 &= \omega_1(s) \\
\dot{\sigma}_2 &= \omega_2(s,\varepsilon)\\
\dot{s} &= 0.
\end{align*}
which means that its solutions are $\big(\sigma_1 + \omega_1(s) t,
\,\, \sigma_2 + \omega_2(s, \varepsilon) t, \,\, s \big)$ on the
space $\Big(\mathbb{R}^2 / 2\pi\mathbb{Z}^2\Big) \times (L,\infty)$.
Since by construction and due to the $\varepsilon$-freedom $\omega_2(s,\varepsilon)/\omega_1(s)$ is not constant, we can
conclude that on infinitely many foliating tori $\text{T}_s$ the
orbits of $\tilde{X}_{\varepsilon}$ exhibit the dynamics of a
Kroneker foliation so that generically, the Kroneker dynamics on
two such tori is non-equivalent.
