We study the equation ν₁(x) = x, where ν₁(x) is the function of frequency of the digit 1 in the ternary expansion of x. We prove that this equation has a unique rational root and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about the equations ν i (x) = x, i = 0, 2, are given.