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tokens: From the ∼56% remaining, 70% of them have a counterpart in either 1RXS or 2RXS catalogs. In this paper, we focus on and perform an extensive visual inspection and literature search of the other 30% of the systems. Of these objects, we describe in detail the analysis and characterization of a pilot sample of 13 candidates. We determined a spectroscopic redshift for each cluster candidate finding that most of them are local systems ( z ≲ 0.05), and are very extended (significant X-ray emission traced out to more than 10 arcmin radius). Moreover, all these objects have fluxes above the limit of the 1RXS and 2RXS catalogs. We argue that they might have been missed because of the source finder used in previous works, that is, the sliding-cell algorithm, which is not optimal for finding such low-flux, very extended objects. Our pilot sample also reveals that the surface-brightness distribution of the systems is flatter than the usual profile, that is β 2/3, and that the missed systems are located in the group regime given the determined masses from galaxy velocity dispersions or a L X − M scaling relation ( M 500 ≲ 10 14 M ⊙ ). In a subsequent paper, we will present and discuss in detail the remaining cluster candidates that we have found. Appendix A: Additional figure Fig. A.1. From left to right : RASS photon images ( left ), optical images, integrated count rates profiles, and spectroscopic redshift histograms ( right ) for the galaxy groups in our pilot sample. The optical images are from ATLAS (composed of g , r , i bands, group 2, 6, 10), SDSS (composed of g , r , i bands, group 1, 3, 7, 8, 9) or DSS ( r band, other groups). The images have a 1 × 1 deg 2 size, except for the ATLAS images, which are 0.30 × 0.30 deg 2 . The red cross shows the position of the group candidate. The magenta plus signs display the location of 1RXS or 2RXS sources. The cyan diamonds show the positions of galaxies used for redshift determination. The white asterisks show the positions of known optical galaxy groups/clusters obtained from NED. The green contours are obtained from the corresponding wavelet-filtered images, and the blue circle shows the extent value (i.e., the core radius) of the group. The integrated count rates profile is background subtracted (solid line). The dashed curves give the 1 σ statistical error of the count rates. The vertical and horizontal dot-dashed lines indicate the aperture radius and the count rates used for the primary photometric results. The dotted curve shows the β model of the cluster modelling used to extrapolate to r 500 (see Sect. 4.3.2 for further details). In the spectroscopic redshift histograms the redshift values are taken for all galaxies within a box of 1 × 1 deg 2 centered on the group position with z 0.10 from NED. The total number of galaxies used in each redshift determination is given by the second number displayed in the upper right corner of each panel (see also Table 3 ). The redshift distribution is fitted