Analogue of Newton–Puiseux series for non-holonomic $D$-modules and factoring

We introduce a concept of a fractional derivatives series and prove that any linear partial differential equation in two independent variables has a fractional derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i.e., non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.