  $\SOL$ geometry is one of the eight homogeneous Thurston 3-geomet-ri-es
$$\EUC, \SPH, \HYP, \SXR, \HXR, \SLR, \NIL, \SOL.$$ In \cite{Sz10} the {\it
densest lattice-like translation ball packings} to a type (type {\bf I/1} in
this paper) of $\SOL$ lattices has been determined. Some basic concept of
$\SOL$ were defined by {\sc{P. Scott}} in \cite{S}, in general.
  In our present work we shall classify $\SOL$ lattices in an algorithmic way
into 17 (seventeen) types, in analogy of the 14 Bravais types of the Euclidean
3-lattices, but infinitely many $\SOL$ affine equivalence classes, in each
type. Then the discrete isometry groups of compact fundamental domain
(crystallographic groups) can also be classified into infinitely many classes
but finitely many types, left to other publication. To this we shall study
relations between $\SOL$ lattices and lattices of the pseudoeuclidean (or here
rather called Minkowskian) plane \cite{AQ}. Moreover, we introduce the notion
of $\SOL$ parallelepiped to every lattice type. From our new results we
emphasize Theorems 3-4-5-6.
  In this paper we shall use the affine model of $\SOL$ space through
affine-projective homogeneous coordinates \cite{M97} which gives a unified way
of investigating and visualizing homogeneous spaces, in general.
