Abstract Let D be a bounded domain in R 2 with a smooth boundary S , N be the unit normal to S pointing out of D , k > 0 is a constant. The class of over-determined problems of the type: ∇ 2 u + k 2 u = c 0 i n D , u | S = c 1 , u N | S = c 2 , is studied. Here u N | S is the normal derivative of u on S and c j , j = 0 , 1 , 2 , are constants. This problem was not studied as far as the author knows. Our result is the following theorem. Theorem. If | c 1 − c 0 k 2 | + | c 2 | > 0 and the above problem has a solution then D is a ball. This theorem contains several earlier results, proved by the author, including the refined Schiffer’s conjecture and the refined Pompeiu problem. This paper continues the author’s investigations of symmetry problems for partial differential equations.