  We show that the recently proposed equations for holomorphic sector of
higher-spin theory in $d=4$, also known as chiral, can be naturally extended to
describe interacting symmetric higher-spin gauge fields in any dimension. This
is achieved with the aid of Vasiliev's off shell higher-spin algebra. The
latter contains ideal associated to traces that has to be factored out in order
to set the equations on shell. To identify the ideal in interactions we observe
the global $sp(2)$ that underlies it to all orders. The $sp(2)$ field dependent
generators are found in closed form and appear to be remarkably simple. The
traceful higher-spin vertices are analyzed against locality and shown to be
all-order space-time spin-local in the gauge sector, as well as spin-local in
the Weyl sector. The vertices are found manifestly in the form of curious
integrals over hypersimplices. We also extend to any $d$ the earlier observed
in $d=4$ higher-spin shift symmetry known to be tightly related to
spin-locality.
