peaks in the nucleon spectra of K$^-$ absorption in $^4$He, [@KEK04; @KEK05]. Additional evidence was given by the FINUDA measurement of the invariant mass distribution of the $\Lambda p $ produced in K$^-$absorption by light nuclei [@FIN05]. The existence of such bound states have been expected as the kaon-nucleon and the kaon-nucleus interactions have been known to be strongly attractive, [@WYC86]. This is now firmly confirmed on the basis of kaonic atom data [@FRI07]. However, the KEK and FINUDA experiments indicate unexpectedly strong bindings of the order of 100, 150 MeV in the lightest KNN, KNNN systems. These experiments require further confirmation. Also, the interpretation of the observed peaks has been disputed in Refs. [@VAL06A], [@VAL06B] while the initial interpretation is defended in Ref. [@AKA07].

Calculations indicate that such states are expected, albeit these might be very broad and difficult to detect. The first calculations performed by Akaishi and Yamazaki in Ref. [@AKA02] were followed by several subsequent publications. These calculations exploited essentially the S - wave resonant attraction related to the $\Lambda(1405)$ state. With an optical model type of approach it was shown that the K-meson optical potential at the center of small nuclei may be as strong as 