Integrability Conditions and Lax Pair of Linear Systems

Non-linear partial differential equations are notoriously difficult to find analytical solutions for. By finding a set of so-called Lax pairs that are linear equations structure of solutions can be revealed. Linearised versions of general relativity and other theories of gravity are used to descibe low gravity regimes like our solar system. Many adaptions of General relativity have followed in the grand traditions of Brans-Dicke-Jordan with their additional scalar field to describe a variable inertial-gravitational interaction strength through space (and time). Linearised version sit at the heart of Penrose's twistor theory as well. It might be interesting to consider the coupled Rarita-Schwinger type equation describing spin 3/2 particles as the Lax pairs associated to these extended gravity theories: such as Galileons and Massive gravity. This was done by Buchdahl for Einstien's equations and Tod for Einstein-Maxwell in which the contct form that arises from Frobenius's theorem and the foliation of space is to be re-iterpreted as a spinor-valued one form.

These extended gravity theories invoke the Vainstein Mechanism (a non-linear kinetic term in the scalar field of a scalar-tensor theory of gravity) to hide the extra degrees of freedom that introducing the scalar field necessarily brings. Given this Mechanism is said to operate beyond a certain radius of (say) our Sun so that non-perturbative (non-liearisable) regmes of General relativity are stitched to gether with linearisable regimes of non-GR theories of gravity.