I am interested in the following linear PDE:

$(1+ t\partial_t) ~ (\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x)A(x,t)+ {t\over (1-x)^3} A(x,t)=0$ 

with $A(x,0)=1$.

This PDE is invariant under the transformations $x \to {x\over x-1},~ t\to -t$, which fix the ratio $(1,6,6)$ in the x-derivative part. 

This PDE looks like a Fuchsian-type equation, but (I think) normally Fuchsian equations assume only one variable – Does anyone know this type of "generalised Fuchsian-type equation" with two variables? I wonder if this is known and studied in the math literature.  

I am interested in the PDE's analytic solutions, especially solutions that are non-perturbative in t, and other symmetry properties.