The chief constable of Greater Manchester Police Peter Fahy has pleaded guilty on behalf of the force to a criminal charge following the death of an officer.
PC Ian Terry, 32, died after being shot in the chest by a blank round during a training exercise in June 2008.
The father of two from Burnley, Lancashire, had not been wearing body armour as the unit practised in a disused factory in Newton Heath.
The officer's widow Joanne was at Liverpool Crown Court to hear the chief constable admit "systemic failure" by the police in Manchester.
The case was brought by the Health and Safety Executive after the Crown Prosecution Service decided not to press charges.
An inquest in March 2010 found that PC Terry was unlawfully killed.
The Greater Manchester force will be sentenced in March this year. The court heard that Mr Fahy was not with the Manchester force during the time of the officer's death but accepted "organisational failures".
Two police training officers identified only by the pseudonyms 'Francis' and 'Eric' pleaded not guilty to breaches of Health and Safety law.
They were granted anonymity and made their pleas from behind a specially erected screen.
Their case has been adjourned to a trial date in June.