Due to Clairin and Goursat, a Bäcklund transformation of the first kind can be associated with Monge-Ampère’s equation. We shall consider Monge-Ampère’s equation of the form s + f(x, y, z, p, q) + g(x, y, z, p, q) t = 0, where p = ∂z/∂x, q = ∂z/∂y, s = ∂2z/∂x∂y, t = ∂2z/∂y2
. The following theorems will be obtained:
1. The transformed equation takes on the same form s′ + f′ + g′t′ = 0 if and only if the given equation can be transformed to a Teixeira equation s + L(x, y, z, q)t + M(x, y, z, q)p + N(x, y, z, q) = 0 by a contact transformation.
2. Teixeira equation s + tL + pM + N = 0 is solved by integrable systems of order n if and only if the transformed equation is solved by integrable systems of order n — 1.