We study restrictions on scattering amplitudes on the worldvolume of branes and strings (such as confining flux tubes in QCD) implied by the target space Poincare symmetry. We focus on exploring the conditions for the string worldsheet theory to be integrable. Even though for higher dimensional membrane the scattering amplitudes for the translational Goldstone modes are double soft, this property is generically violated for the strings. Violation of double softness implies in turn the breakdown of integrability. We prove that if Goldstones are the only gapless degrees of freedom, integrability can only be present for D = 3 or 26 . We extend the analysis to include massless worldsheet fermions, resulting from spontaneous breakdown of the target space supersymmetry. In this case the tree-level integrability is in one-to-one correspondence with the existence of a κ-symmetric Green-Schwarz action.