Catalytic branching random walk (CBRW) describes reproduction of particles and their movement in space. The particles may give offspring in the presence of catalysts. Consider CBRW model where the particles perform a random walk over a multidimensional lattice, without procreation outside the catalysts. The latter take a finite number of fixed positions at the lattice. We study the particles spread in case of non-extinction of population initiated by a single particle. The rate of population spread depends essentially on the distribution tails of the random walk jumps. Consider the jumps with independent (or close to independent) components having regularly varying “heavy’’ tails. The main results show that, after a proper normalization of positions, in the time limit the particles concentrate on a random set, located at the coordinate axes. For a two-dimensional case, the limiting set forms a cross, and, for any higher dimension d, it is a collection of d segments containing the origin. The joint distribution of such segments lengths is found and the time-limit is understood in the sense of weak convergence. This radically differs from the known results for both the CBRW with “light” and semi-exponential distribution tails of the random walk jumps.