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Bug introduced in 8.0.4 or earlier, persisting through 12.0.

DSolve quickly returns solutions to the following PDE (which is the homogeneous portion of the PDE in question 130755).

s = Flatten@DSolve[D[l[w1, w2], w1] a w2 - D[l[w1, w2], w2] a w1 - l[w1, w2] == 0, 
    l[w1, w2], {w1, w2}]
(* {l[w1, w2] -> E^(-(ArcTan[w1/Sqrt[w2^2]]/a)) C[1][1/2 (w1^2 + w2^2)], 
    l[w1, w2] -> E^(ArcTan[w1/Sqrt[w2^2]]/a) C[1][1/2 (w1^2 + w2^2)]} *)

However, an attempt to verify this result indicates that one of the two solutions is spurious.

FullSimplify[Unevaluated[D[l[w1, w2], w1] a w2 - D[l[w1, w2], w2] a w1 - l[w1, w2]] /. #] &
    /@ s
(* {-((E^(-(ArcTan[w1/Sqrt[w2^2]]/a)) (w2 + Sqrt[w2^2]) C[1][1/2 (w1^2 + w2^2)])/w2), 
      (E^(ArcTan[w1/Sqrt[w2^2]]/a) (-w2 + Sqrt[w2^2]) C[1][1/2 (w1^2 + w2^2)])/w2} *)

The first term fails to vanish for w2 > 0, and the second term for w2 < 0. Executing SetOptions[Solve, Method -> Reduce] prior to DSolve in the hope of obtaining conditional answers produces the same result. Also, using the DSolve Assumptions option does not help. For instance,

sp = Flatten@DSolve[D[l[w1, w2], w1] a w2 - D[l[w1, w2], w2] a w1 - l[w1, w2] == 0, 
    l[w1, w2], {w1, w2}, Assumptions -> w2 > 0]
(* {l[w1, w2] -> E^(-(ArcTan[w1/w2]/a)) C[1][1/2 (w1^2 + w2^2)], 
    l[w1, w2] -> E^(ArcTan[w1/w2]/a) C[1][1/2 (w1^2 + w2^2)]} *)

FullSimplify[Unevaluated[D[l[w1, w2], w1] a w2 - D[l[w1, w2], w2] a w1 - l[w1, w2]] /. #] &
    /@ sp
(* {-2 E^(-(ArcTan[w1/w2]/a)) C[1][1/2 (w1^2 + w2^2)], 0} *)

Once again, one solution is spurious. In fact, the correct solution is

l[w1, w2] -> E^(-(ArcTan[w1, w2]/a)) C[1][1/2 (w1^2 + w2^2)]];
FullSimplify[Unevaluated[D[l[w1, w2], w1] a w2 - D[l[w1, w2], w2] a w1 - l[w1, w2]] /. %
(* 0 *)

My questions are, (1) is this a bug (as it appears to be)?, and (2) does a work-around exist (apart from changing independent variables to obtain an ODE instead)?

Addendum

As commented by xzczd, this problem also occurs in question 130596.

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