The integro-differential equations d²ⁿ/dx²ⁿ ∫_-1 ¹ (a[(x - t)²] ln |x - t| + b[(x - t)²])φ(t)dt = f(x) of the convolution on an interval with infinitely differentiable functions a(s) and b(s) decreasing at infinity are considered. The Fourier symbol is assumed to be sectorial, that is, it has positive projection on some direction in the complex plane. The existence and uniqueness of solutions in the classes of functions representable in the form φ(t) = (1 - t²)^δn φ(t), δ_n = n - 1 + ε, ε < 0, φ ∈ C¹ [-1, 1] are proved. Properties concerning the smoothness of solutions are described.