  We present all-dielectric polaritonic metasurfaces consisting of properly
sculptured cylinders to sustain the dynamic anapole, i.e. a non-radiating
alternating current distribution. One way for the anapole to emerge, is by
combining modes based on the first and the fourth Mie resonance of a cylinder
made of high permittivity LiTaO$_3$ for operation in the low THz. The circular
cross-section of each cylinder varies periodically along its length in a binary
way, from small to large, while its overall circular symmetry has to be broken
in order to remove parasitic magnetic modes. Small cross-sections are the main
source of the \textit{electric dipole} Mie mode, while large cross-sections
sustain the fourth, \textit{mixed toroidal dipole} Mie mode. With proper
adjustment, the generated electric and toroidal moments interfere destructively
producing a non-radiating source, the dynamic anapole, the existence of which
is attested by a sharp dip in the reflection from the metasurface, due
exclusively to electric and toroidal dipoles. Moreover, we show that, by
breaking the circular symmetry of each cylinder, a substantial toroidal dipole
emerges from the \textit{magnetic quadrupole} Mie mode, which in combination
with the electric dipole also produces the dynamic anapole. The sensitivity of
the anapole states to the material dissipation losses is examined leading to
the conclusion that the proposed metasurfaces offer a scheme for realistically
implementing the anapole.
