A new method for investigation of the ultra-heavy component of Galactic cosmic-ray nuclei (Z = 50-92) has been developed since 1980. It depends on the ability of extraterrestrial silicate crystals (olivine) to register and store during many million years the tracks due to cosmic-ray nuclei with Z~22. Our approach .bases on the partial annealing of both "fossil" tracks and the tracks due to accelerated Kr, Xe, Au, Pb and U ions, and on the chemical etching of total volume track length of the cosmic-ray nuclei in the olivine crystals. The crystals taken from Marjalahti and Eagle Station pallasites (the radiation ages are 180 and 45 million years respectively) were annealed at 703 K during 32 h and etched. The volume tracks due to Z5 0 cosmic-ray nuclei were then measured. The intercomparison of track length spectra of the "fossil" tracks and the tracks. due to the accelerated 238U and 208Pb nuclei proves unambiguously that the last two abundant peaks of "fossil" track length spectra (120-140 J.lm and 180 J.lm, first observed in 1980 in Dubna) are products of the recent nucleosynthesis in our Galaxy due to Pt-Pb and Th-U groups of cosmic-ray nuclei. During 1980-1996 more than 1600 Th-U "fossil" tracks were measured and 11 anomalously long tracks (L = 340-370 J.lm) were found. Concerning the origin of the anomalously long tracks we have determined the orientation of these tracks in olivine crystal with the Laue-Roentgen method. We estabilished that five anomalously long "fossil" tracks have an angle 2:: 15°to the main crystal plane (010) and subsequently can not be attributed to any actinide nuclei in meteoritic olivine crystals. Moreover, in the fossil track study of olivines annealed at 703 K during 32 h we found the 250 J.lm-long track which could not be attributed to Th-U nuclei at any orientation in the crystal lattice. Now we can set an upper limit ofZ~110 (superheavy) nuclei abundance at the level N z~l lo/N Th , u::;; (173).10-3. Our goal is to obtain the finat necessary and sufficient proofs of the existence of SHE nuclei in Galactic matter.