Context : It has recently been discovered that the Galactic bulge is X-shaped , with the two southern arms of the X both crossing the lines of sight at l = 0 and |b| > 4 , hence producing a double red clump in the bulge color magnitude diagram .
Dynamical models predict the formation of X-shaped bulges , as extreme cases of boxy-peanut bulges .
However , since X-shaped bulges were known to be present only in external galaxies , models have never been compared to kinematical data for individual stars .

Aims : We study the orbital motion of Galactic bulge stars , in the two arms ( overdensities ) of the X in the southern hemisphere .
The goal is to provide observational constraints to bulge formation models that predict the formation of X-shapes through bar dynamical instabilities .

Methods : Radial velocities have been obtained for a sample of 454 bulge giants , roughly equally distributed between the bright and faint red clump , in a field at ( l,b ) = ( 0 , -6 ) .
Proper motions were derived for all red clump stars in the same field by combining images from two epochs obtained 11 years apart , with WFI @ 2.2m at La Silla .
The observed field contains the Globular Cluster NGC 6558 , whose member stars were used to assess the accuracy of the proper motion measurement .
At the same time , as a by product , we provide the first proper motion measurement of NGC 6558 .
The proper motions for the spectroscopic sub-sample are analyzed taking into account the radial velocities and metallicities measured from near-IR Calcium triplet lines , for a subsample of 352 stars .

Results : The radial velocity distribution of stars in the bright red clump , tracing the closer overdensity of bulge stars , shows an excess of stars moving towards the Sun .
Similarly , an excess of stars receding from the Sun is seen in the far overdensity , traced by faint red clump stars .
This is explained with the presence of stars on elongated orbits , most likely streaming along the arms of the X-shaped bulge .
Proper motions for these stars are consistent with qualitative predictions of dynamical models of peanut–shaped bulges .
Surprisingly , stars on elongated orbits have preferentially metal poor ( subsolar ) metallicities , while the metal rich ones , in both overdensities , are preferentially found in more axysimmetric orbits .

The observed proper motion of NGC 6558 has been measured as ( \mu _ { l } \cos ( b ) , \mu _ { b } ) = ( 0.30 \pm 0.14 , -0.43 \pm 0.13 ) , with a dispersion of ( \sigma _ { l } \cos ( b ) , \sigma _ { b } ) = ( 1.8 , 1.7 ) mas/yr .
Being the first PM measurement for this cluster .

Conclusions :