I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables, $x=(x_1,...,x_n)$ (using multi-index notation), $ P(x,D)=\sum_{\alpha}^m P_\alpha(x)D^\alpha$ with polynomial coefficients, $P_\alpha(x)$, what are the holonomic modules that can be associated to this operator?
I know that for $n=1$ every such operator defines a holonomic D-module, so is there a simple algorithm, for example, in $n=2$ that would determine immediately if the operator is holonomic or not?
A single PDE in $n > 1$ variables cannot be holonomic. The characteristic variety of a D-module defined by one PDE is just the vanishing locus of its principal symbol, considered as a function on the cotangent bundle. A D-module is holonomic when its characteristic variety is Lagrangian, or equivalently has dimension equal to its codimension.
In this case, the cotangent bundle is $\mathbb{R}^{2n}$ and the principal symbol is $$P(x,y) = \sum_{\alpha}^m P_{\alpha}(x)y^{\alpha},$$ where $y_1,\cdots,y_n$ are new variables. The vanishing locus of $P(x,y)$ has codimension $1$, so for $n > 1$ cannot have (co)dimension $n$, which is the holonomicity condition.