Context :

Aims : We characterize the importance of metallicity on the presence of molecular hydrogen in damped Lyman- \alpha ( DLA ) systems .

Methods : We construct a representative sample of 18 DLA/sub-DLA systems with log N ( H i ) > 19.5 at high redshift ( z _ { abs } > 1.8 ) with metallicities relative to solar [ X/H ] > - 1.3 ( with [ X/H ] = log N ( X ) / N ( H ) - log ( X/H ) _ { \odot } and X either Zn , S or Si ) .
We gather data covering the expected wavelength range of redshifted H _ { 2 } absorption lines on all systems in the sample from either the literature ( 10 DLAs ) , the UVES-archive or new VLT-UVES observations for four of them .
The sample is large enough to discuss for the first time the importance of metallicity as a criterion for the presence of molecular hydrogen in the neutral phase at high- z .

Results : From the new observations , we report two new detections of molecular hydrogen in the systems at z _ { abs } = 2.431 toward Q 2343 + 125 and z _ { abs } = 2.426 toward Q 2348 - 011 .
We compare the H _ { 2 } detection fraction in the high-metallicity sample with the detection fraction in the overall sample from Ledoux et al .
( 2003 ) .
We show that the fraction of DLA systems with log f = log 2 N ( H _ { 2 } ) / ( 2 N ( H _ { 2 } ) + N ( H i ) ) > - 4 is as large as 50 % for [ X/H ] > - 0.7 when it is only \sim 5 % for [ X/H ] < - 1.3 and \sim 15 % in the overall sample ( with - 2.5 < [ X/H ] < - 0.3 ) .
This demonstrates that the presence of molecular hydrogen at high redshift is strongly correlated with metallicity .

Conclusions :