Defining relations and identities of finite-generated nilpotent algebra $R$ with condition $\dim R^{N}/R^{N+1} = 2$

In this paper we describe defining relations of $s$-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 degree $N+2$ if $s\geq N$, or the standard identity of smaller degree than $N$ if $s < N$.