Self-similar reductions of the Sawada-Kotera (1) and Kupershmidt (2) equations are studied in the paper \[y_{zzzzz}+\frac{5}{2}nyy_{xxx}+\frac{5}{2}ny_{x}y_{xx}+\frac{5}{4}n^{2}y^{2}y_{x}-2y-zy_{z}=0\tag{1}\] \[w_{zzzzz}+10nww_{xxx}+25nw_{xx}+20n^{2}w^{2}w_{x}-2w-zw_{z}=0\tag{2}\] \[y=y(z),w=w(z),z=x(5t)^{-\frac{1}{5}}.\] The author shows that these equations pass the Painlevé test and he finds the Lax pair for them. To find exact solutions to the above equations, the author proceeds as follows. The differential polynomial \(P\) on the left hand side of the equation is represented as a superposition (product) \(P=L\circ Q\) of the linear polynomial \(L\) and the non-linear polynomial \(Q\), the zeros of which have already been studied. For example, for the first equation, this representation looks like this \((n=1)\) \[y_{zzzzz}+\frac{5}{2}yy_{xxx}+\frac{5}{2}yy_{xx}+\frac{5}{4}yy_{x}-2y-zy_{z}=(\frac{d^{^{3}}}{dz^{3}}+2y\frac{d}{dz}+y_{z})(y_{zz}+\frac{1}{4}y^{2}-z).\] Since the zeros of the polynomial \(Q\equiv y_{zz}+\frac{1}{4}y^{2}-z\) are expressed in terms of the solutions of the first Pelevé equation, then the latters represent solutions of equation (1) too. In a similar way, the author proves that equations (1) and (2) have solutions representable by means of general solution for the first member of \(K_{2}\) hierarchy. Rational solutions of the equations (1) and (2) also are given in the paper. Special polynomials are introduced for their expression.