  Considering the importance of ever-increasing interest in exploring localized
waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito
equation describing the unidirectional propagation of shallow-water waves and
perform Painlev\'e analysis to understand its integrability nature. We
construct the explicit form of higher-order rogue wave solutions by adopting
Hirota's bilinearization and generalized polynomial functions. Further, we
explore their dynamics in detail, depicting different pattern formation that
reveal potential advantages with available arbitrary constants in their
manipulation mechanism. Particularly, we demonstrate the existence of
singly-localized line-rogue waves and doubly-localized rogue waves with
multiple (single, triple, and sextuple) structures generating triangular and
pentagon type geometrical patterns with controllable orientations that can be
altered appropriately by tuning the parameters. The presented analysis will be
an essential inclusion in the context of rogue waves in higher-dimensional
systems.
