Homogeneous freedom in a scalar singular integral equation ($w=\tilde{w}(C\rho +H)$)

For the scalar jump $(a+b)w_{+}+(b-a)w_{-}=c$, divide by a nonvanishing canonical homogeneous solution $\tilde{w}$ to obtain an additive jump $\rho =c/((a+b){\tilde{w}}_{+})$. Its Cauchy integral gives a particular quotient, while every additional quotient $H$ with no jump is homogeneous freedom. The density $f=w_{+}-w_{-}$ therefore differs from the normalized particular density by $-2b{\tilde{w}}_{+}H/(a-b)$. Normalization, index and endpoint conditions determine the allowable $H$; dropping it without such conditions does not give the general solution.