  Let $G$ be a group. Associate a graph $\mathcal{E}_G$ (called the co-Engel
graph of $G$) with $G$ whose vertex set is $G$ and two distinct vertices $x$
and $y$ are adjacent if $[x, {}_k y] \neq 1$ and $[y, {}_k x] \neq 1$ for all
positive integer $k$. This graph, under the name ``Engel graph'', was
introduced by Abdollahi. Let $L(G)$ be the set of all left Engel elements of
$G$. In this paper, we realize the induced subgraph of co-Engel graphs of
certain finite non-Engel groups $G$ induced by $G \setminus L(G)$. We write
$\mathcal{E}^-(G)$ to denote the subgraph of $\mathcal{E}_G$ induced by $G
\setminus L(G)$. We also compute genus, various spectra, energies and Zagreb
indices of $\mathcal{E}^-(G)$ for those groups. As a consequence, we determine
(up to isomorphism) all finite non-Engel group $G$ such that the clique number
is at most $4$ and $\mathcal{E}^-$ is toroidal or projective. Further, we show
that $\coeng{G}$ is super integral and satisfies the E-LE conjecture and the
Hansen--Vuki{\v{c}}evi{\'c} conjecture for the groups considered in this paper.
