Let $M$ be a riemannian manifold.
Recall that a Singular Riemannian Foliation (SRF) $\mathcal{F}$ on $M$ is defined as a partition of $M$ into submanifolds of $M$ immersed injectively (called leaves) $\{ L_i\}$ such that there exists vector fields $\{X_i\}$ in $M$ with $T_pL_p$ is generated by $\{X_i(p)\}$ for each $p \in M$ and if $\gamma$ is a geodesic in $M$ and $\gamma \perp L_x$ for some $x \in M$, then $\gamma \perp L_p$ for all $p \in M$. (Here, consider $L_p$ as the leaf containing $p \in M$)
If $\mathcal{F}$ is a SRF on $M$, the dual foliation of $\mathcal{F}$ is defined as the family of dual leaves, where the dual leaf to a leaf $L_x \in \mathcal{F}$ is defined as
$$ L^\#_x=\{ q\in M~|~\exists c:[0,1]\to M,~c(0)=x,~c(1)=q,~\dot{c}(t)\perp T_{c(t)}L~\forall t\in[0,1]\}$$
Is there an example of a dual foliation of a SRF that is not a SRF?