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We construct families of symmetric, antisymmetric, and asymmetric solitarymodes in one-dimensional bichromatic lattices with thesecond-harmonic-generating ($\chi ^{(2)}$) nonlinearity concentrated at a pairof sites placed at distance $l$. The lattice can be built as an array ofoptical waveguides. Solutions are obtained in an implicit analytical form,which is made explicit in the case of adjacent nonlinear sites, $l=1$. Thestability is analyzed through the computation of eigenvalues for smallperturbations, and verified by direct simulations. In the cascading limit,which corresponds to large mismatch $q$, the system becomes tantamount to therecently studied single-component lattice with two embedded sites carrying thecubic nonlinearity. The modes undergo qualitative changes with the variation of$q$. In particular, at $l\geq 2$, the symmetry-breaking bifurcation (SBB),which creates asymmetric states from symmetric ones, is supercritical andsubcritical for small and large values of $q$, respectively, while thebifurcation is always supercritical at $l=1$. In the experiment, thecorresponding change of the phase transition between the second and first kindsmay be implemented by varying the mismatch, via the wavelength of the inputbeam. The existence threshold (minimum total power) for the symmetric modesvanishes exactly at $q=0$, which suggests a possibility to create the solitarymode using low-power beams. The stability of solution families also changeswith $q$.

Symmetric and asymmetric localized modes in linear lattices with an embedded pair of $\chi ^{(2)}$-nonlinear sites