Consider the following linear PDE: $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ jontly with the initial value condition $A(x,0)=1$.

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 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.

Note added:

For a slightly simpler PDE $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ t A(x,t)=0 $$ with $A(x,0)=1$, I found that the exact solution is given by a hypergeometric ${_0F_3}$ function: $$ A(x,t)=\;{_0F_3}\left(~;{{4\over 3}, {5\over 3},2}; -{x^3 t\over 27}\right). $$ This I believe corresponds to the small $x$ limit of the exact solution to the first PDE with the factor ${1\over (1-x)^3}$. (It took me several pages of computation to get the above solution so if anyone can re-derive this solution in a few lines I'd be very happy to know!) In any case, I wonder if there is a method that might help obtain the exact solution to the first PDE.