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We use the method of equivariant differential geometry to prove the existence of a complete hypersurface with constant negative scalar curvature in      E n ( n ≥  4 ) {E^n}(n \geq 4)      . This is the first example of a complete hypersurface with constant negative scalar curvature in      E n ( n ≥  4 ) {E^n}(n \geq 4)      .

${\rm O}(2)\times{\rm O}(2)$-invariant hypersurfaces with constant negative scalar curvature in $E\sp 4$