 Solution

The initial data force $C=0$, so the implicit relation factors as $\sin y(x-\sin y)=0$. Near $(0,0)$ this permits either the constant branch or the sine branch:Both satisfy the original equation, including its degenerate value $0=0$ at the initial point. They have different slopes there, zero and one, so the initial value problem is not unique. There are no further differentiable local branches through that point: away from $x=0$ the two factors cannot switch, and joining the two different branches at zero would give unequal one-sided derivatives.

The nonconstant branch is defined as a solution for $-1<x<1$; the constant branch exists for every real $x$.