How about we write it as just
By inspection you can see is a solution including the solution right? So letting and making that substitution, I get and I can do the division to obtain only if and . Doing that, and integrating twice, I obtain the solution:
. Now note that the solutions are not particular cases of this solution and it looks like the general solutions are asymptotically tangent to the solution but not the other y=k solutions. I'm a little unsure about this part as to whether all the y=k solutions are singular or just the y=0 one.