Let $A=\mathbb{C}[x,y,u,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of http://mathoverflow.net/questions/26651/is-there-a-simple-method-to-test-a-local-ring-to-be-cohen-macaulay, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free. 

My questions is: how to compute $M$ concretely?(say as a quotient of sum of $A$)

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilda(M)$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

Thanks for the help!