| Summary: | In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>A</mi> <mrow> <mn>5</mn> <mo>,</mo> <mn>7</mn> </mrow> <mrow> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mrow> </msubsup> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>A</mi> <mn>18</mn> <mi>a</mi> </msubsup> </semantics> </math> </inline-formula>. For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.
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