  We have explored the hidden symmetries of a generic four-parameter
nearest-neighbor spin model, allowed in honeycomb lattice compounds under
trigonal compression. Our method utilizes a systematic algorithm to identify
all dual transformations of the model that map the Hamiltonian on itself,
changing the parameters and providing exact links between different points in
its parameter space. We have found the complete set of points of hidden SU(2)
symmetry at which seemingly highly anisotropic model can be mapped back on the
Heisenberg model and inherits therefore its properties such as the presence of
gapless Goldstone modes. The procedure used to search for the hidden symmetries
is quite general and may be extended to other bond-anisotropic spin models and
other lattices, such as the triangular, kagome, hyper-honeycomb, or
harmonic-honeycomb lattices. We apply our findings to the honeycomb lattice
iridates Na2IrO3 and Li2IrO3, and illustrate how they help to identify
plausible values of the model parameters that are compatible with the available
experimental data.
