Flat subspaces of the $SL(n,\mathbb{R})$ chiral equations

In this work, we introduce a method for finding exact solutions to the vacuum Einstein field equations in higher dimensions from a given solution to the chiral equation. When considering a $n + 2$-dimensional spacetime with $n$ commutative Killing vectors, the metric tensor can take the form $\hat g = f ( \rho, \zeta ) ( d \rho^2 + d \zeta^2 ) + g_{\mu \nu} ( \rho, \zeta ) d x^\mu d x^\nu$. Then, the Einstein field equations in vacuum reduce to a chiral equation, $( \rho g_{, z} g ^{-1} )_{, \bar z} + ( \rho g_{, \bar z} g ^{-1} )_{, z} = 0$, and two differential equations, $( \ln f \rho ^{1-1/n} )_{, Z} = \frac{\rho}{2} \operatorname{tr} ( g_{, _Z} g^{-1} )^2$, where $g \in SL( n, \mathbb{R} )$ is the normalized matrix representation of $g_{\mu \nu}$, $z = \rho + i \zeta$ and $Z = z, \bar z$. We use the ansatz $g = g ( \xi^a )$, where the parameters $\xi^a$ depend on $z$ and $\bar z$ and satisfy a generalized Laplace equation, $( \rho \xi^a _{, z} )_{, \bar z} + ( \rho \xi^a _{, \bar z} )_{, z} = 0$. The chiral equation to the Killing equation, $A_{a , \xi^b} + A_{b , \xi^a} = 0$, where $A_a = g_{, \xi^a} g^{-1}$. Furthermore, we assume that the matrices $A_a$ commute with each other; in this way, they fulfill the Killing equation.