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We present a definition of mutations of species with potential that can beapplied to the species realizations of any skew-symmetrizable matrix B overcyclic Galois extensions E/F whose base field F has a primitive [E:F]-th rootof unity. After providing an example of a globally unfoldableskew-symmetrizable matrix whose species realizations do not admitnon-degenerate potentials, we present a construction that associates a specieswith potential to each tagged triangulation of a surface with marked points andorbifold points of order 2. Then we prove that for any two taggedtriangulations related by a flip, the associated species with potential arerelated by the corresponding mutation (up to a possible change of sign at acycle), thus showing that these species with potential are non-degenerate. Inthe absence of orbifold points, the constructions and results specialize toprevious work by the second author.  The species constructed here for each triangulation $\tau$ is a speciesrealization of one of the several matrices that Felikson-Shapiro-Tumarkin haveassociated to $\tau$, namely, the one that in their setting arises fromchoosing the number 1/2 for every orbifold point.

Species with potential arising from surfaces with orbifold points of order 2, Part I: one choice of weights