For a heavy enough Higgs-boson field, the Weinberg-Salam theory admits classical solutions additional to the sphaleron. These are called deformed sphalerons. They induce nontrivial modifications in the space of the classical configurations of this spontaneously broken gauge theory. We construct paths which extrapolate between the vacuum, the sphaleron, and the deformed sphalerons. The space of finite-energy configurations around the sphaleron is also investigated by means of the eigenvectors with negative eigenvalues associated with this extremum.