<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Z</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> -balls: Solitons from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Z</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> -symmetric scalar field theory

We discuss the conditions under which static, finite-energy, configurations of a complex scalar field $\ensuremath{\phi}$ with constant phase and spherically-symmetric norm exist in potential form $V({\ensuremath{\phi}}^{*}\ensuremath{\phi},{\ensuremath{\phi}}^{N}+{\ensuremath{\phi}}^{*N})$ $N\ensuremath{\in}\mathbb{N}$ $N\ensuremath{\ge}2$, i.e., ${Z}_{N}$-symmetry. Such are called ${Z}_{N}$-balls. build explicit solutions ($3+1$)-dimensions from model mimicking effective theories based on Polyakov loop finite-temperature $\mathrm{SU}(N)$ Yang-Mills theory. find ${Z}_{N}$-balls for $N=3$, 4, 6, 8, 10, show that only static zero radial nodes $N$ odd, while may even.