Let the region S={( x, y)∥ μ( x+ iy, x− iy)>0} be the interior of Steiner's hypocycloid, where μ(z, z)=−z 2 z 2+4z 3+4 z 3−18z z+27 . For each real α>− 5 6 an orthogonal system of polynomials p m,n α(z, z), m, n⩾0 , can be defined on this region S such that p m,n α(z, z)−z m z n has degree less than m+ n and ∫∫ S p α m,n(z, z) q(z, z) (μ(z, z)) α dx dy=0 for each polynomial q of degree less than m+ n. If z=e i(s+ t √3 ) +e i(−s+ t √3 ) + e − 2it √3 then, in terms of s and t, the functions p m,n − 1 2 and μ 1 2 p m−1,n−1 1 2 are the regular eigenfunctions of the operator ∂ 2 ∂s 2 + ∂ 2 ∂t 2 which remain invariant or change sign, respectively, under the reflections in the edges of a certain equilateral triangle. Two explicit partial differential operators D 1 α and D 2 α in z and z of orders two and three, respectively, are obtained such that the polynomials p m, n α are eigenfunctions of D 1 α and D 2 α . The operators D 1 α and D 2 α commute and are algebraically independent, and they generate the algebra of all differential operators for which the polynomials p m, n α are eigenfunctions. If α=0, 1 2 , 3 2 or 7 2 then the operator D 1 α expressed in terms of s and t is the radial part of the Laplace-Beltrami operator on certain compact Riemannian symmetric spaces of rank two.