Ultrasound measurements have revealed a remarkable lattice softening of the transverse elastic modulus (C11 - C12)/2 in single-crystalline La2 - xSrxCuO4 only around x = 1/8. Analysis of the temperature dependence of the elastic modulus indicates that in the particular range of carrier concentration there exists a narrow electronic band in the vicinity of the Fermi level which couples to the shearing strain εxx - εyy.