$\mathbb Z_3$-orthograded quasimonocomposition algebras with one-dimensional null component

We consider $\mathbb Z_3$-orthograded nondegenerate quasimonocomposition algebras $A=A_0\oplus A_1\oplus A_2$ such that $\dim A_0=1$ and $A_1A_2=0$. It is proved that all algebras in this class $W$ are solvable of solvability index either two or three. All non bi-isotropic orthogonal nonisomorphic algebras $A$ of $W$ of least dimension, which is equal to $9$, are classified. An infinite series of algebras $C_r$ in $W$ of dimension $\dim C_r=8r+1$ is constructed for every $r\in\mathbb N=\{1,2,\dots\}$. All algebras $C_r$ are solvable of solvability index $3$ and nilpotent of nil-index $5$.