It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential equation of the second order to a canonical form throughout a domain. I read the above result, for example, in Renuka Ravindran's book, Partial Differential Equations, Page 58~59. Since I have not found any detailed explanation of this result, I want to construct a counterexample of PDE in at least three independent variables, like this $$\sum_{i,j=1}^n a_{ij}(x)u_{x_ix_j}+\sum_{i=1}^n b_{i}(x)u_{x_i}+c(x)u(x)=0, x\in\Omega\subset\mathbb{R}^n, $$ where $n\geq3,$ such that which can not be reduced to canonical form globally, that is the form of the following $$\sum_{i,j=1}^n \alpha_{ij} u_{\xi_i\xi_j}+\text{lower oder terms}=0, ~ \xi\in\Omega'\subset\mathbb{R}^n,$$ where $$\alpha_{i,j}=0,\pm 1.$$ But I do not how. Can anyone help me, or tell me where I can find these counterexamples?