  We comprehensively study admissible transformations between normal linear
systems of second-order ordinary differential equations with an arbitrary
number of dependent variables under several appropriate gauges of the arbitrary
elements parameterizing these systems. For each class from the constructed
chain of nested gauged classes of such systems, we single out its singular
subclass, which appears to consist of systems being similar to the elementary
(free particle) system whereas the regular subclass is the complement of the
singular one. This allows us to exhaustively describe the equivalence groupoids
of the above classes as well as of their singular and regular subclasses.
Applying various algebraic techniques, we establish principal properties of Lie
symmetries of the systems under consideration and outline ways for completely
classifying these symmetries. In particular, we compute the sharp lower and
upper bounds for the dimensions of the maximal Lie invariance algebras
possessed by systems from each of the above classes and subclasses. We also
show how equivalence transformations and Lie symmetries can be used for
reduction of order of such systems and their integration. As an illustrative
example of using the theory developed, we solve the complete group
classification problems for all these classes in the case of two dependent
variables.
