On a conservative integral equation with two kernels

We study the solvability of the integral equation
$$f(x)=g(x)+\int_0^\infty T_1(x-t)f(t)\,dt+\int_{-\infty}^0T_2(x-t)f(t)\,dt,\qquad x\in\mathbb R,$$
where $f\in L_1^{\operatorname{loc}}(\mathbb R)$ is the unknown function and $g$, $T_1$ and $T_2$ are given functions satisfying the conditions
$$g\in L_1(\mathbb R),\quad 0\le T_j\in L_1(\mathbb R),\quad \int_{-\infty}^\infty T_j(t)\,dt=1,\qquad j=1,2.$$
Most attention is paid to the nontrivial solvability of the homogeneous equation
$$s(x)=\int_0^\infty T_1(x-t)s(t)\,dt+\int_{-\infty}^0T_2(x-t)s(t)\,dt,\qquad x\in\mathbb R.$$