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We show that the integral method used to prove the no-hair theorem for BlackHoles (BH's) in spherically symmetric and static spacetimes within theframework of general relativity with matter composed by a complex-valuedscalar-field does not lead to a straightforward conclusion about the absence ofhair in the stationary and rotating (axisymmetric) scenario. We argue that sucha failure can be used to justify in a simple and heuristic way the existence ofnon-trivial boson clouds or hair found numerically by Herdeiro and Radu [1,2]and analytically by Hod in the test field limit [3-5]. This is due to thepresence of a contribution that is negative when rotation exists which allowsfor an integral to vanish even when a non-trivial boson hair is present. Thepresence of such a negative contribution that depends on the rotationproperties of the BH is perfectly correlated with the eigenvalue problemassociated with the boson-field equation. Conversely, when the rotation isabsent the integral turns to be composed only by non negative (i.e. positivesemidefinite) terms and thus the only way it can vanish is when the hair iscompletely absent. This analysis poses serious challenges and obstructionstowards the elaboration of no-hair theorems for more general spacetimes endowedwith a BH region even when including matter fields that obey the energyconditions. Thus rotating boson stars, if collapsed, may lead indeed to a newtype of rotating BH, like the ones found in [1,2]. In order to achieve thisanalysis we solve numerically the eigenvalue problem for the boson field in theKerr-BH background by imposing rigorous regularity conditions at the BH horizonfor the non-extremal case ($0<a<M$) which include the near extremal one in thelimit $M\rightarrow a$, as well as the small BH limit $M\rightarrowa\rightarrow 0$.

Obstructions towards a generalization of no-hair theorems: I. Scalar clouds around Kerr black holes