Let $$W\subset \mathbb {P}^{13}$$
W
⊂
P
13
be the image of the rational map defined by the linear system of the sextic surfaces of $$\mathbb {P}^3$$
P
3
having double points along the edges of a tetrahedron. Let $$\mathcal {L}$$
L
be the linear system of the hyperplane sections of W. It is known that a general $$S\in \mathcal {L}$$
S
∈
L
is an Enriques surface. The aim of this paper is to study the sublinear system $$\mathcal {L}_{\bullet }\subset \mathcal {L}$$
L
∙
⊂
L
of the hyperplane sections of W having a triple point at a general point $$w \in W$$
w
∈
W
. We will show that a general element of $$\mathcal {L}_{\bullet }$$
L
∙
is birational to an elliptic ruled surface and that the image of W via the rational map defined by $$\mathcal {L}_{\bullet }$$
L
∙
is a cubic Del Pezzo surface $$\Delta \subset \mathbb {P}^3$$
Δ
⊂
P
3
with 4 nodes. Interestingly, this fact appears to be related to a conjecture of Castelnuovo.