I have been studying exceptional sheaves and their mutations on Del Pezzo surfaces (specifically, on $\mathbb{P}^1 \times \mathbb{P}^1$). Given an exceptional pair $(E,F)$ of sheaves there are a priori three different kinds of mutations that can occur:

1. "regular/division": when $E\otimes \text{Hom}(E,F) \twoheadrightarrow F$

2. "rebound/recoil": when $E\otimes \text{Hom}(E,F) \hookrightarrow F$

3. "extension": when $\text{Ext}^1(E,F) \neq 0$

However, I have never seen an example of the second "rebound/recoil" type. Drezet proves (see note after Theorem 6) that these do not exist on $\mathbb{P}^2$. Is there an example of these on another Del Pezzo surface that I have missed, or is there a conjecture stating that none exist?

Also, if an example does exist, does there exist an example where $(E,F)$ is in a thread of a helix?

Notes:

-A sheaf on a surface is exceptional if $\text{Ext}^\star(E,E) = \text{End}(E,E) = \mathbb{C}$.

-$(E,F)$ is an exceptional pair if $\text{Ext}^\star(F,E)=0$ and $\text{Ext}^i(E,F)$ is non-zero for at most one $i$.

-the left mutation of $F$ though $E$, denoted $L_E F$, is defined by the triangle $L \longrightarrow E\otimes \text{Hom}(E,F) \stackrel{\text{eval}}{\longrightarrow} F \longrightarrow L[1]$$L \longrightarrow E\otimes \text{Hom}^\bullet(E,F) \stackrel{\text{eval}}{\longrightarrow} F \longrightarrow L[1]$, and either $L$ or $L[1]$ is chosen as the mutation depending on if the mutation is of type 1,2, or 3. Note: it is also possible to define $L_F E$ always as $L$ or always as $L[1]$