This paper is motivated by a general question: for which values of k and n is the universal Burnside kei Q̅(k,n) finite? It is known (starting from the work of M. Takasaki (1942)) that Q̅(2,n) is isomorphic to the dihedral quandle Zₙ and Q̅(3,3) is isomorphic to Z₃ ⊕ Z₃. In this paper, we give a description of the algebraic structure for Burnside keis Q̅(4,3) and Q̅(3,4). We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation a = ⋯ a∗b∗ ⋯ ∗a∗b. Invariants of links related to the Burnside kei Q̅(k,n) are invariant under n-moves.