Infinitesimal generator Similarity variables Reduction equation ${X}_{1}$ $\varsigma =t,$ $u\left(x,t\right)=F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$ ${F}^{″}=0,$ ${H}^{″}=0.$ ${X}_{2}$ $\varsigma =x,$ $u\left(x,t\right)=F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$ $F{F}^{\prime }+{H}^{\prime }=0,$ ${F}^{\prime }+H{F}^{\prime }+F{H}^{\prime }+{F}^{\left(3\right)}=0.$ (D) ${X}_{3}$ $\varsigma =t,$ $u\left(x,t\right)=x/t+F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$ $F+\varsigma {F}^{\prime }=0,$ $H+\varsigma {H}^{\prime }+1=0.$ ${X}_{4}$ $\varsigma =x{t}^{-1/2},$ $u\left(x,t\right)={t}^{-1/2}F\left(\varsigma \right),v\left(x,t\right)=-1+{t}^{-1}H\left(\varsigma \right).$ $F+\varsigma {F}^{\prime }-2F{F}^{\prime }-2{H}^{\prime }=0,$ $2H-2H{F}^{\prime }+\varsigma {H}^{\prime }-2F{H}^{\prime }-2{F}^{\left(3\right)}=0.$ (A) ${X}_{1}+{X}_{2}$ $\varsigma =x-t,$ $u\left(x,t\right)=F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$ $-{F}^{\prime }+F{F}^{\prime }+{H}^{\prime }=0,$ ${F}^{\prime }+H{F}^{\prime }-{H}^{\prime }+F{H}^{\prime }+{F}^{\left(3\right)}=0.$ (B) ${X}_{2}+{X}_{3}$ $\varsigma =-{t}^{2}+2x/2,$ $u\left(x,t\right)=t+F\left(\varsigma \right),v\left(x,t\right)=H\left(\varsigma \right).$ $1+F{F}^{\prime }+{H}^{\prime }=0,$ ${F}^{\prime }+H{F}^{\prime }+F{H}^{\prime }+{F}^{\left(3\right)}=0.$ (C)