theory of extremal rays [@mo2], and such pairs $(X,\mathbb{E})$ are classified in the cases $r=n+1$, $n$, and $n-1$ by Ye-Zhang [@yz], Peternell [@p0] [@p], Fujita [@fn], and Peternell-Szurek-Wiśniewski [@psw]. Note here that the projective space bundle $\mathbb{P}(\mathbb{E})$ is a Fano manifold; in such a case, $\mathbb{E}$ is called a Fano bundle. In this vein, in view of $-K_{\mathbb{P}(\mathbb{E})}=rH(\mathbb{E})$, $\mathbb{E}$ might be called a “Del-Pezzo bundle” if $r=n-1$, and a “Mukai bundle” if $r=n-2$. As is well known, Mukai manifolds, i.e., Fano manifolds of coindex three, are described in [@MR0995400], whereas the “Mukai bundle” $\mathbb{E}$ has not been investigated for an arbitrary rank $r$ even if the underlying manifold $X$ is simple such as $\mathbb{P}^n$ or $\mathbb{Q}^n$. Thus the present deepest result in this direction is the classification of “Del-Pezzo bundles” due to Peternell-Szurek-Wiśniewski [@psw]. Roughly speaking, their method of classification is to relate the pair $(X,\mathbb{E})$ with a pair $(X,\mathcal{E})$ by setting $\mathcal{E}=\mathbb{E}(-1)$, and show that $\mathcal{E}$ is nef by their Comparison lemma [@psw (3.1)]: an ample $\mathbb{E}$ can be replaced by a nef $\mathcal{E}$. Then they classified such $(X,\mathcal{E})$’s in [@pswnef] in the following cases: if $X$ is isomorphic to a projective $\mathbb{P}^n$ and the first Chern class $c_1(\mathcal{E})$ 