Some nonlinear equations reducible to diffusion-type equations (Q1592153)

The problem under consideration is: what conditions on \(\mathbf A\) and \(\mathbf\Gamma\) ensure the existence of some pointwise transformation \(\widetilde{y}^i = \widetilde{y}^i(y^1,\ldots, y^n)\), \(i=1,\ldots,n\), locally reducing the system \[ \frac{\partial y^i}{\partial t} = \sum^n_{j=1}A^i_j\left(\frac{\partial^2 y^j}{\partial x^2}+ \sum^n_{r=1}\sum^n_{s=1}\Gamma^i_{rs} \frac{\partial y^r}{\partial x}\frac{\partial y^s}{\partial x} \right) \] to the diffusion type?