We study the Gibbons-Tsarev equation z(yy) + z(x)z(xy) - z(y)z(xx) + 1 = 0 and, using the known Lax pair, we construct infinite series of conservation laws and the algebra of nonlocal symmetries in the covering associated with these conservation laws. We prove that the algebra is isomorphic to the Witt algebra. Finally, we show that the constructed symmetries are unique in the class of polynomial ones.