Structure, defining relations and identities of finite-dimensional nilpotent algebra $R$ with condition $dim\, R^{N}/R^{N+1} = 2$

In this paper we describe structure and defining relations of $2$-generated nilpotent algebra $R$ over arbitrary field with condition $dim R^{N}/R^{N+1} = 2$ for some natural number $N \geq 3$. It is proved that such algebra $R$ over a field of characteristic not two satisfies the standard identity of much smaller degree than $N$ (for large values of $N$).