  In this paper we describe some Leibniz algebras whose corresponding Lie
algebra is four-dimensional Diamond Lie algebra $\mathfrak{D}$ and the ideal
generated by the squares of elements (further denoted by $I$) is a right
$\mathfrak{D}$-module. Using description \cite{Cas} of representations of
algebra $\mathfrak{D}$ in $\mathfrak{sl}(3,{\mathbb{C}})$ and
$\mathfrak{sp}(4,{\mathbb{F}})$ where ${\mathbb{F}}={\mathbb{R}}$ or
${\mathbb{C}}$ we obtain the classification of above mentioned Leibniz
algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended
to the case of the algebra $\mathfrak{D}.$ Classification of Leibniz algebras
with corresponding Lie algebra $\mathfrak{D}$ and with the ideal $I$ as a Fock
right $\mathfrak{D}$-module is presented. The linear integrable deformations in
terms of the second cohomology groups of obtained finite-dimensional Leibniz
algebras are described. Two computer programs in Mathematica 10 which help to
calculate for a given Leibniz algebra the general form of elements of spaces
$BL^2$ and $ZL^2$ are constructed, as well.
