where lambda > 0 is a small parameter, vertical bar S vertical bar is the area of S, Delta(g) is the Laplace-Beltrami operator and dv(g) is the area element. Given any integer k >= 1, under general conditions on S we find a bubbling solution u(lambda), which blows up at exactly k points in S, as lambda -> 0. When S is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for k = 2. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If S is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with k = 1 exists. (C) 2018 Elsevier Inc. All rights reserved.