  We point out that the Kitaev materials may not necessarily support Kitaev
spin liquid. It is well-known that having a Kitaev term in the spin interaction
is not the sufficient condition for the Kitaev spin liquid ground state. Many
other spin liquids may be stabilized by the competing spin interactions of the
systems. We thus explore the possibilities of non-Kitaev spin liquids in the
honeycomb Kitaev materials. We carry out a systematic classification of gapped
$\mathbb{Z}_2$ spin liquids using the Schwinger boson representation for the
spin variables. The presence of strong spin-orbit coupling in the Kitaev
materials brings new ingredients into the projective symmetry group
classification of the non-Kitaev spin liquid. We predict the spectroscopic
properties of these gapped non-Kitaev spin liquids. Moreover, among the gapped
spin liquids that we discover, we identify the spin liquid whose spinon
condensation leads to the magnetic Bragg peak structure of the zig-zag magnetic
order that was observed in Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$. We further
discuss the possibility of gapped $\mathbb{Z}_2$ spin liquid and the deconfined
quantum criticality from the zig-zag magnetic order to spin dimerization in
pressurized $\alpha$-RuCl$_3$.
