When one extends the (almost) collapsible pseudo-spine representation theorem for homotopy \(3\)-spheres [Po3] to open simply connected \(3\)-manifolds \(V^3\), new phenomena appear: at the source of the representation, the set of double points is, generally speaking, no longer closed. We show that at the cost of replacing \(V^3\) by \(V_h^3 = \{ V^3 \text{ with very many holes}\}\), we can always find representations \(X^2 \stackrel{f}{\rightarrow} V^3\) with \(X^2\) locally finite and almost-arborescent, with \(\Psi (f)=\Phi (f)\), with the open regular neighbourhood (the only one which is well-defined here) Nbd\((fX^2)=V^3_h\) and such that on any precompact tight transversal to the set of double lines, we have only finitely many limit points (of the set of double points). Moreover, if \(V^3\) is the universal covering space of a closed \(3\)-manifold, \(V^3=\widetilde M^3\), then we can find an \(X^2\) with a free \(\pi_1M^3\) action and having the equivariance property \(f(gx)=gf(x)\), \(g\in \pi_1M^3\). Having simultaneously all these properties for \(X^2\stackrel{f}{\rightarrow} \widetilde M^3\) is one of the steps in the first author's program for proving that \(\pi_1^\infty \widetilde M^3=~0\), [Po11, Po12]. Achieving equivariance is far from being straightforward, since \(X^2\) is gotten starting from a tree of fundamental domains on which \(\pi_1M^3\) cannot, generally speaking, act freely. So, in this paper we have both a representation theorem for general (\(\pi_1=0\)) \(V^3\)'s and a harder equivariant representation theorem for \(\widetilde M^3\) (with \(gfX^2=fX^2, \, g\in\pi_1M^3\)), the proof of which is not a specialization of the first, “easier” result. But, finiteness is achieved in both contexts. In a certain sense, this finiteness is a best possible result, since if the set of limit points in question is \(\emptyset\) (i.e. if the set of double points is closed), then \(\pi_1^\infty V_h^3\) (which is always equal to \(\pi_1^\infty V^3\) ) is zero. In [PoTa2] it was also shown that when we insist on representing \(V^3\) itself, rather than \(V_h^3\), and if \(V^3\) is wild (\(\pi_1^\infty\not =0\)), then the transversal structure of the set of double lines can exhibit chaotic dynamical behavior. Our finiteness theorem avoids chaos at the cost of a lot of redundancy (the same double point \((x, y)\) can be reached in many distinct ways starting from the singularities).