We present new spectroscopic data in the field of five high-redshift (z>=0.6) candidate galaxy clusters, drawn from the EIS Cluster Candidate Catalog. A total of 327 spectra were obtained using FORS1 at the VLT, out of which 266 are galaxies with secure redshifts. In this paper, we use these data for confirming the existence of overdensities in redshift space at the approximate same location as the matched-filter detections in the projected distribution of galaxies from the EIS I-band imaging survey. The spectroscopic redshifts, associated to these overdensities, are consistent but, in general, somewhat lower than those predicted by the matched-filter technique. Combining the systems presented here with those analyzed earlier, we have spectroscopically confirmed a total of nine overdensities in the redshift range 0.6<z<1.3, providing an important first step in building an optically-selected, high-redshift sample for more detailed studies, complementing those based on the few available X-ray selected systems.
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