Suppose $k\in L^2([0,1]\times[0,1])$ is a symmetric kernel. Presumably the following is true(under the assumption that $k$ is weakly singular):
Either $$\displaystyle\int_0^1f(y)k(x,y)dy=0$$or
$$\displaystyle\int_0^1f(y)k(x,y)dy=1$$ has a non-trivial solution. In some papers (for instance in here)this result is attributed to Muskhelishvili and I.Vekua. Unfortunately the right edition of the book(s) is hard to find and the edition I have access to, does not contain such a result.