Let G be a linear algebraic group over an algebraically closed field k and Autalg(G) the group of all algebraic group automorphisms of G. For every φ∈Autalg(G) let R(φ) denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by (g,x)↦gxφ(g−1), for all g,x∈G). We say that G has the algebraic R∞-property if R(φ) is infinite for every φ∈Autalg(G). In [1] we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group G has the algebraic R∞-property, then Gφ (the fixed-point subgroup of G under φ) is infinite for all φ∈Autalg(G). In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic R∞-property and identify certain classes of solvable algebraic groups for which the property fails.