In a related post: Algorithm for solving binary quadratic Diophantine equations (BQDE) and its CTC there was a conclusion that with regard to general binary quadratic diophantine equation (with all non-zero coefficients), the solutions (if existent) can be found in exponential time.
However, is this also the case for simple hyperbolic case (i.e. where $A$ and $C$ are zero, leading to form: $Bxy + Dx + Ey + F = 0$)? Alpertron (https://www.alpertron.com.ar/METHODS.HTM) shows a method for such a case which involves finding all integer divisors of $DE-BF$ - a task which can be tricky for large coefficients. For certain, sufficiently large coefficients Alpertron's method would imply that this can be computed in sub-exponential time (using GNFS).
Is there any algorithm to be used for SHCDE which would find the solutions in less than exponential time (unlike standard ones for general BQDE)?