The course of the Battle at Lake Regillus reveals the strict military discipline demanded of Romans from a very early date and finalized the rejection of the kings.
At the beginning of the 5th century B.C., shortly after the expulsion of the Etruscan kings, the Romans claim to have won a battle at Lake Regillus. Livy describes it in Book II of his history of Rome from its founding.
The battle, which, like most events of the period, contains legendary elements, was part of a war between Rome and a coalition of Latin states, often called the Latin League.
The war ended in 493 B.C. with the Treaty of Spurius Cassius, which was inscribed on a bronze pillar in the Roman Forum where Cicero was still able to provide eye witness testimony to it four centuries later. The treaty established perpetual peace and a defensive military alliance between the two sides, Rome and the Latin League.
At the Lake Regillus Battle, the Romans were led by a dictator, A. Postumius Albus, with Titus Aebutius as master of the horse, and helped by the mythological twins, the Dioscuri, Castor and Pollux, on horseback. For their help, following the victory, the Romans dedicated a temple to them in 484 B.C.
Regillus was a particularly fierce battle in this war, according to Livy, because there were Tarquins, including the last king of Rome, the expelled Tarquinius Superbus, on the Latin side. This angered the Romans and led the generals to participate in the actual combat.
The Latin side, led by Octavus Mamilius, included some Romans who had been exiled at the start of the Republic. The Latin side was fighting so well that Romans started fleeing. Seeing the turn for the worse, the Roman dictator told his (still fresh) cohort to treat any Roman who fled the battle as an enemy. The weary Roman soldiers, re-motivated, went back into the battle and (perhaps) won, although the greater war would not end for another 3 or more years. The exact timing depends on whether Regillus was actually fought in 499, 496, or at all.
Cary and Scullard in A History of Rome say that the alliance of Latins that had helped Rome defeat the Etruscans, after the expulsion of the Tarquins, was turned against Rome at Lake Regillus. Lake Regillus was in the territory of Tusculum, which had probably become the leader of the league.
Source: T.J. Cornell, The Beginnings of Rome