  Spin-orbital liquids are quantum disordered states in systems with entangled
spin and orbital degrees of freedom. We study exactly solvable spin-orbital
models in two dimensions with selected Heisenberg-, Kitaev-, and $\Gamma$-type
interactions, as well as external magnetic fields. These models realize a
variety of spin-orbital-liquid phases featuring dispersing Majorana fermions
with Fermi surfaces, nodal Dirac or quadratic band touching points, or full
gaps. In particular, we show that Zeeman magnetic fields can stabilize
nontrivial flux patterns and induce metamagnetic transitions between states
with different topological character. Solvable nearest-neighbor biquadratic
spin-orbital perturbations can be tuned to stabilize zero-energy flat bands. We
discuss in detail the examples of $\mathrm{SO}(2)$- and
$\mathrm{SO}(3)$-symmetric spin-orbital models on the square and honeycomb
lattices, and use group-theoretical arguments to generalize to
$\mathrm{SO}(\nu)$-symmetric models with arbitrary integer $\nu > 1$. These
results extend the list of exactly solvable models with spin-orbital-liquid
ground states and highlight the intriguing general features of such exotic
phases. Our models are thus excellent starting points for more realistic
modellings of candidate materials.
