  The most precise realization of inertial reference frame in astronomy is the
catalogue of 212 defining extragalactic radiosources with coordinates obtained
during VLBI observation runs in 1979-1995. IAU decided on the development of
the second realization of the ICRF2 catalogue. The criteria of best sources
selection (in terms of coordinates stability) must be defined as the first aim.
The selected sources have to keep stable the coordinate axes of inertial
astronomical frame.
  Here we propose new criteria of source selection for the new ICRF catalogue.
The first one we call as "cosmological" and the second one as "kinematical".
The physical basis of these criteria is based on the assumption that apparent
motion of quasars (at angular scale of the order of hundred microarcseconds) is
connected with real motion inside quasars. Therefore apparent angular motion
corresponds to real physical motion of a "hot spot" inside a radio source. It
is shown that interval of redshift $0.8 \div 3.0$ is the most favorable in
terms that physical shift inside such sources corresponds to minimal apparent
angular shift of a "hot spot". Among "cosmologically" selected sources we
propose to select motionless sources and sources with linear motion which are
predictable and stable over long time interval.
  To select sources which satisfies such conditions we analyzed known redshifts
of sources and time series obtained by our code and by different centers of
analysis of VLBI data. As a result of these analyses we select 137 sources as a
basis for the ICRF2 catalogue.
