Question: Calculating representations of matrix group and point (Lie) symmetries of differential operators.

Hello,

I would like to see if MAPLE fits our needs. In order to see that I would like to consider the example which is posted in H. Iwaniec :"Introduction to the spectral theory (...)" p. 36, that is as follows:

Suppose that you have a PDE as follows:

This PDE has the following point symmetries:

These differential operators are the representations of the following matrix algebra:

1. I would like to see the MAPLE commands, which give the algebra $v_1, v_2, v_3$ 

2. I would like to see those commands, which give you the algebra $v_1, v_2, v_3$ as a representation of $X_1, X_2, X_3$.

3. Given the algebra $v$ I would like to see the commands which give you the Laplace-Beltrami operator $\Delta_{LB}=y\partial_{xx} + y\partial_{yy}$

( P.s. sorry for the intransparent form, I do not know how to insert formulae in the comment.)

Thanks in advance.

G45073