  We target multivariable series associated with resolutions of complex
analytic normal surface singularities. In general, the equivariant
multivariable analytical and topological Poincar\'e series are well-defined and
have good properties only if the link is a rational homology sphere. We wish to
create a model when this assumption is not valid: we analyse the case of cusps.
For such germs we define even the motivic versions of these two series, we
prove that they are equal, and we provide explicit combinatorial expression for
them. This is done via a motivic multivariable series associated with the space
of effective Cartier divisors of the reduced exceptional curve.
