Flat pencils of metrics and integrable reductions of Lamé's equations (Q2784564)

The author proposes a method of solving a particular system of PDE's which is obtained by reduction of so called Lamé equations related to the problem of compatibility of flat pseudo-Riemannian metrics. (Two such metrics \(g_1\) and \(g_2\) are said to be compatible iff the Levi-Civita connections and curvature tensors of any linear combination \(a_1g_1 + a_2g_2\) with constant coefficients \(a_i\) are of the form \(a_1\nabla_1 + a_2\nabla_2\) and \(a_1R_1 + a_2R_2\), \(\nabla_i\) and \(R_i\) being the Levi-Civita connections and curvature tensors of \(g_i\), \(i = 1, 2\)).