The author employs exterior differential systems to investigate the generalized soliton equation \({\mathcal L}_X g= -2 \text{Ric} g+g\) with the trace requirement \(\Sigma g^{ij} \Gamma^k_{ij} =0\) (to ensure the ellipticity in the Douglis-Nirenberg sense) for Riemannian metrics \(g\). The involutivity test is thoroughly analyzed and gives the result that, modulo diffeomorphisms, the \(n\)-dimensional soliton metrics locally depend on \(n(n+1)\) functions on \(n-1\) variables.