$]{}]{} occurs iff $ \Z_\mu = {\left\{ 0 \right\}} $.\
[(ii)]{} The case [[$ ({\bf C2}) $]{}]{} occurs iff $ \Z_\mu = \Z $. If it is true, then, among the solutions, all extremal points are strong and the others non-strong.\
[(iii)]{} The case [[$ ({\bf C3}) $]{}]{} occurs iff $ {\left\{ 0 \right\}} \subsetneq \Z_\mu \subsetneq \Z $.

This result sounds somehow paradoxical; at least, they are mysterious: [*Uniqueness in law implies that the solution is non-strong, and existence of strong solution(s) implies that it is non-unique but extremal*]{}.

The purpose of the present paper is to generalize Yor’s results. We study the following stochastic equation on a compact group $ G $ in discrete negative time: $$\eta_k = \eta_{k-1} \cdot \xi_k , 
\quad k \in -\N, 
\label{STYE}$$ which we call the [*simple Tsirelson–Yor equation on $ G $*]{}. We shall introduce the notions of a solution, a strong solution and uniqueness in law following the theory of stochastic differential equations. Then we obtain the following result, which generalizes Theorems 2 of [@Yor].

\[Intro exist\] For any given noise law $ \mu $, there exists a solution $ \bP^*_\mu $ of the equation such that all marginal distributions of 