Isometric immersions of $S^2$ to $S^4$

Question

There are non-standard isometric immersions of the sphere $S^2(r)$ of radius $0<r<4$ into the unit sphere $S^4$. Such immersion have been constructed in [1]. As shown in [1], these immersions are actually isometric embeddings when $0<r<1.6$. In particular we have a non-standard isometric embedding of the unit sphere $S^2$ into $S^4$.

Is there an isometric immersion of the unit sphere $S^2$ into $S^4$ which is not an embedding (has self-intersections)?

[1] D. Ferus, U. Pinkall, Constant curvature 2-spheres in the 4-sphere. Math. Z. 200 (1989), 265-271.

EDIT: The Veronese embedding mentioned by j.c. in his comment is described here.

EDIT2: The examples show that there are isometric immersions of $S^2(r)$ into the unit sphere $S^4$ that have self-intersections for $r<1$ and for some $r>1$. The original question about the case $r=1$ is still not answered.

Answer 1

Here is a simple source of such immersions for $r<1$.

Consider geodesic subspheres $\mathbb{S}^2\subset \mathbb{S}^3\subset \mathbb{S}^4$. Take a closed curve $\gamma$ in $\mathbb{S}^2$. Pass to its spherical suspension; it is embedded in $\mathbb{S}^3$. Pass to the spherical suspension again; you get an immersed singular surface $\Sigma^3$ in $\mathbb{S}^4$ which admits an immersion of hemisphere $$\iota\colon\mathbb{S}^3_+\looparrowright\Sigma^3.$$

Note that for $r<1$, there are isometric embeddings $f\colon\mathbb{S}^2(r)\to \mathbb{S}^3_+$ and for the right choice of $f$ and $\gamma$, the composition $\iota\circ f\colon\mathbb{S}^2(r)\to \mathbb{S}^4$ has self-intersections.