The paper deals with a non-classical boundary value problem of linear plane elasticity. With the aid of singular integral equations, Fredholm’s theorems are proved. Using complex variable techniques, the problem is considered for the unit circle and epitrochoids. It is proved that for every positive integer n and within every arbitrarily small vicinity of the unit circle there exists such an epitrochoid for which the homogeneous non-classical problem allows at least n linearly independent solutions.