  We have studied the $P\to\gamma^\star\gamma^\star$ form factor in Resonance
Chiral Theory, with $P = \pi^0,\eta,\eta'$, to compute the contribution of the
pseudoscalar pole to the hadronic light-by-light piece of the anomalous
magnetic moment of the muon. In this work we allow the leading $U(3)$ chiral
symmetry breaking terms, obtaining the most general expression for the form
factor of order $\mathcal{O}(m_P^2)$. The parameters of the Effective Field
Theory are obtained by means of short distance constraints on the form factor
and matching with the expected behavior from QCD. Those parameters that cannot
be fixed in this way are fitted to experimental determinations of the form
factor within the spacelike momentum region of the virtual photon. Chiral
symmetry relations among the transition form factors for $\pi^0,\eta$ and
$\eta'$ allow for a simultaneous fit to experimental data for the three mesons.
This shows an inconsistency between the BaBar $\pi^0$ data and the rest of the
experimental inputs. Thus, we find a total pseudoscalar pole contribution of
$a_\mu^{P,HLbL}=(8.47\pm 0.16)\cdot 10^{-10}$ for our best fit (neglecting the
BaBar $\pi^0$ data). Also, a preliminary rough estimate of the impact of NLO in
$1/N_C$ corrections and higher vector multiplets (asym) enlarges the
uncertainty up to $a_\mu^{P,HLbL}=(8.47\pm 0.16_{\rm stat}\pm
0.09_{N_C}{}^{+0.5}_{-0.0_{\rm asym}})$
