This article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged O(3) sigma model,(E)−Δu+A0(∏j=1k|x−pj|2nj)−aeu(1+eu)1+a=4π∑j=1knjδpj−4π∑j=1lmjδqjinR2. In this equation the {δpj}j=1k (resp. {δqj}j=1l) are Dirac masses concentrated at the points {pj}j=1k, (resp. {qj}j=1l), nj and mj are positive integers, and a is a nonnegative real number. We set N=∑j=1knj and M=∑j=1lmj.In previous works [7,24], some qualitative properties of solutions of (E) with a=0 have been established. Our aim in this article is to study the more general case where a>0. The additional difficulties of this case come from the fact that the nonlinearity is no longer monotone and the data are signed measures. As a consequence we cannot anymore construct directly the solutions by the monotonicity method combined with the supersolutions and subsolutions technique. Instead we develop a new and self-contained approach which enables us to emphasize the role played by the gravitation in the gauged O(3) sigma model. Without the gravitational term, i.e. if a=0, problem (E) has a layer's structure of solutions {uβ}β∈(−2(N−M),−2], where uβ is the unique non-topological solution such that uβ=βln|x|+O(1) for −2(N−M)<β<−2 and u−2=−2ln|x|−2lnln|x|+O(1) at infinity respectively. On the contrary, when a>0, the set of solutions to problem (E) has a much richer structure: besides the topological solutions, there exists a sequence of non-topological solutions in type I, i.e. such that u tends to −∞ at infinity, and of non-topological solutions of type II, which tend to ∞ at infinity. The existence of these types of solutions depends on the values of the parameters N,M,β and on the gravitational interaction associated to a.