We propose non-linear formally consistent equations of motion for the Type-B Higher Spin Gravity that is dual to the free fermion or to the Gross-Neveu model, depending on the boundary conditions. The equations are directly obtained from the first principles: the gauge invariance of the CFT partition function on an arbitrary background. We show that the system has a vacuum solution describing general higher-spin flat backgrounds and demonstrate that the respective linearized system describes propagation of higher-spin fields over such backgrounds, reproducing all the structures that are known to determine nonlinear higher-spin equations.

Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS5 x S5 superstring world-sheet theory in terms of physical degrees of freedom we investigate a Pohlmeyer-reduced version of the corresponding supercoset sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting gauge-fixed equations can be obtained from a Lagrangian of a non-abelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. The final form of the Pohlmeyer-reduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. Its small-fluctuation spectrum is that of 8 bosonic and 8 fermionic degrees of freedom with equal masses. We show that in the special case of the AdS2 x S2 superstring model the reduced theory is supersymmetric: it is equivalent to the (2,2) supersymmetric extension of the sine-Gordon model.