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We study a continuous-time branching random walk on the lattice$\mathbb{Z}^{d}$, $d\in \mathbb{N}$, with a single source of branching, that isthe lattice point where the birth and death of particles can occur. The randomwalk is assumed to be homogeneous, symmetric and irreducible but, in contrastto previous investigations, the random walk transition intensities $a(x,y)$decrease as $|y-x|^{-(d+\alpha)}$ for $|y-x|\to \infty$, where$\alpha\in(0,2)$, that leads to an infinite variance of the random walk jumps.The~mechanism of the birth and death of particles at the source is governed bya continuous-time Bienaym\'e-Galton-Watson branching process. The sourceintensity is characterized by a certain parameter $\beta$. We calculate thelong-time asymptotic behaviour for all integer moments for the number ofparticles at each lattice point and for the total population size. With respectto the parameter $\beta$ a non-trivial critical point $\beta_c>0$ is found forevery $d\geq 1$. In particular, if $\beta>\beta_{c}$ the evolutionary operatorgenerated a behaviour of the first moment for the number of particles has apositive eigenvalue. The existence of a positive eigenvalue yields anexponential growth in $t$ of the particle numbers in the case $\beta>\beta_c$called \emph{supercritical}. Classification of the branching random walktreated as \emph{subcritical} ($\beta<\beta_c$) or \emph{critical}($\beta=\beta_c$) for the heavy-tailed random walk jumps is more complicatedthan for a random walk with a finite variance of jumps. We study the asymptoticbehaviour of all integer moments of a number of particles at any point$y\in\mathbb{Z}^d$ and of the particle population on $\mathbb{Z}^d$ accordingto the ratio $d/\alpha$.

Heavy-Tailed Branching Random Walks on Multidimensional Lattices. A Moment Approach