  This paper gives a complete classification of conics in $PE_2(\mathbb{R})$.
The classification has been made earlier (Reveruk [5]), but it showed to be
incomplete and not possible to cite and use in further studies of properties of
conics, pencil of conics, and of quadratic forms in pseudo-Euclidean spaces.
This paper provides that. A pseudo-orthogonal matrix, pseudo-Euclidean values
of a matrix, diagonalization of a matrix in a pseudo-Euclidean way are
introduced. Conics are divided in families and by types, giving both of them
geometrical meaning. The invariants of a conic with respect to the group of
motions in $PE_2(\mathbb{R})$ are determined, making it possible to determine a
conic without reducing its equation to canonical form. An overview table is
given.
