Context : B [ e ] supergiants are known to have non-spherical winds , and the existence of disks that are neutral in hydrogen close to their stellar surface has been postulated .
A suitable mechanism to produce non-spherical winds seems to be rapid rotation , and at least for three B [ e ] supergiants in the Magellanic Clouds rotation velocities at a substantial fraction of their critical velocity have been found .

Aims : We want to find suitable recombination distances in the equatorial plane of rapidly rotating stars that explain the observed huge amounts of neutral material in the vicinity of B [ e ] supergiants .

Methods : We perform ionization structure calculations in the equatorial plane around rapidly rotating luminous supergiants .
The restriction to the equatorial plane allows us to treat the ionization balance equations 1-dimensionally , while the stellar radiation field is calculated 2-dimensionally , taking into account the latitudinal variation of the stellar surface parameters .
The stellar parameters used correspond to those known for B [ e ] supergiants .
The assumptions made in the computations all have in common that the total number of available ionizing photons at any location within the equatorial plane is overestimated , resulting in upper limits for the recombination distances .

Results : We find that despite the drop in equatorial surface density of rapidly rotating stars ( neglecting effects like bi-stability and/or wind compression ) , hydrogen and helium recombine at or close to the stellar surface , for mass loss rates \dot { M } \ga 5 \times 10 ^ { -5 } M _ { \odot } yr ^ { -1 } and rotation speeds in excess of v _ { rot,eq } / v _ { crit } \simeq 0.8 .

Conclusions :