From the IC:

c= 0

Therfore you would think

x^4 y^2 + x^3 y^4 = 0 is the solution

However

dy/dx = (4x^3 y^2 + 3x^2 y^4)/2x^4 y + 4x^3 y^3)

reduces to:

dy/dx = (4x^3 y + 3x^2 y^3)/2x^4 + 4x^3 y^2)

which has the equiilibrium solution y = 0 as well

Another way of looking at this is

x^4 y^2 + x^3 y^4 = 0

y^2( x^4 +x^3y^2) =0

which yields y = 0 or x = -y^2 as you suggested.

Note

dy/dx = (4x^3 y + 3x^2 y^3)/2x^4 + 4x^3 y^2)

does not satisfy the conditions of the uniqueness theorem.