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Can a billiard map be locally linear near a periodic orbit of period 2? This question can be asked in all dimensions beginning from 2. It is more interesting to consider the situation when the linear symplectic map to which the billiard map is locally conjugated, is totally elliptic. Then the billiard is locally integrable. Answer to this question is still unknown. We reduce the question to analysis of some equations which have formal solutions, but conjugacy of the corresponding series is still unproven. Numerical results suggest a conjecture that such a conjugacy takes place and several other conjectures which will be discussed in the talk.