In this paper we begin the study of <TEX>$L_k$</TEX>-2-type hypersurfaces of a hypersphere <TEX>${\mathbb{S}}^{n+1}{\subset}{\mathbb{R}}^{n+2}$</TEX> for <TEX>$k{\geq}1$</TEX> Let <TEX>${\psi}:M^3{\rightarrow}{\mathbb{S}}^4$</TEX> be an orientable <TEX>$H_k$</TEX>-hypersurface, which is not an open portion of a hypersphere. Then <TEX>$M^3$</TEX> is of <TEX>$L_k$</TEX>-2-type if and only if <TEX>$M^3$</TEX> is a Clifford tori <TEX>${\mathbb{S}}^1(r_1){\times}{\mathbb{S}}^2(r_2)$</TEX>, <TEX>$r^2_1+r^2_2=1$</TEX>, for appropriate radii, or a tube <TEX>$T^r(V^2)$</TEX> of appropriate constant radius r around the Veronese embedding of the real projective plane <TEX>${\mathbb{R}}P^2({\sqrt{3}})$</TEX>.