Police Scotland's chief constable is to stay on in his post despite being the subject of an inquiry over allegations of gross misconduct.
Phil Gormley had faced calls to step aside while he is under investigation by the police watchdog.
BBC Scotland understands the probe concerns an allegation about bullying.
Scottish Lib Dem leader Willie Rennie said it would be "necessary" for Mr Gormley to take a leave of absence to ensure an effective investigation.
However, after a meeting on Thursday afternoon the Scottish Police Authority (SPA) said a suspension was "not appropriate" in the circumstances.
Other parties have called for "full transparency" in the investigation and want the full findings to be published.
The probe follows a referral from the SPA, which passed the matter to the Police Investigations and Review Commissioner (Pirc) for investigation.
There has been no formal indication of the nature of the complaint made against Mr Gormley, or who submitted it. However, Pirc said that "the allegations, if proved, would amount to gross misconduct" - defined as a breach of professional standards under which "dismissal may be justified".
BBC Scotland understands the complaint was one of bullying, from a fellow senior officer at Police Scotland.
Mr Gormley, who took up the post of chief constable in January 2016, said he was "cooperating fully with Pirc" and would provide "all necessary assistance".
But he added: "I would like to stress that I remain focused on leading Police Scotland, ensuring that we continue to serve and protect the people of this country."
Mr Rennie said the allegations were "incredibly serious" and needed "a thorough and prompt investigation".
He said: "You cannot have the chief constable, the boss of the organisation, in place while his office is being investigated.
"I think it would be appropriate for him to step to one side. I think it would be wise if he took that decision himself.
"This is a very serious allegation, about gross misconduct, so it wouldn't be appropriate for the chief constable to remain in position while that investigation is being conducted."
Niven Rennie, a former president of the Association of Police Superintendents, also said it might be best if Mr Gormley stepped aside for the duration of the inquiry.
He told BBC Scotland: "There is the potential that witnesses might not come forward because he's there, or might not say something they otherwise would have said.
"There's also a question about other members of the executive team - they'll be getting interviewed as well. Did they see such conduct? And if they did, why did they not step in to stop it? So the whole matter is very complicated."
Analysis by Reevel Alderson, BBC Scotland home affairs correspondent
BBC Scotland understands the allegations come from a senior officer at Police Scotland headquarters at Tulliallan Castle and are of bullying behaviour.
Normally these investigations would be investigated by the force's professional standards department, but because Mr Gormley is of such a senior rank, the regulations require that it's passed to the Scottish Police Authority.
They don't have any investigation capabilities so they have passed it to the Pirc - and that investigation is now under way.
Mr Gormley has said it is inappropriate to comment further while that is ongoing. In the meantime, he continues to work while he awaits the results of this inquiry.
However, some have said there is no need for Mr Gormley to stand aside while the probe is in process.
Scottish Conservative justice spokesman Liam Kerr said: "I presume what's being said is that he should actually be suspended while the investigation is ongoing. I just don't think that would be terribly helpful.
"We don't know the full nature of the allegation, we don't know the full nature of the investigation. We only know that potentially it might be gross misconduct - but it might not be.
"There's a process of dealing with issues like this, and it needs to be allowed to run its course."
SPA deputy chairwoman Nicola Marchant said the board had met to consider whether suspension was appropriate.
She said: "At this stage, and having carefully considered and balanced the various investigatory and public interest criteria within the regulations, the SPA takes the view that a suspension is not appropriate. As with any process of this nature, that is an issue we will keep under review.
"While complaints and conduct issues relating to senior officers are handled within a clear set of guidelines and procedures, the circumstances behind each case are different and so therefore should any consideration of whether a period of suspension is appropriate."
Mr Kerr said it was "essential" that there was "full transparency as to what exactly has happened".
Scottish Labour's Claire Baker also urged Pirc to be "as transparent as possible", saying: "With the most senior police officer under investigation, it is vital that whatever the outcome the public maintains confidence in Police Scotland."
Scottish Green MSP John Finnie, himself a former policeman, said it was "vital that a thorough investigation is undertaken and the full findings are published".
Ms Marchant said investigations must be allowed to progress confidentially, but said it was in the public interest that information would come out "in due course".
She said: "While we recognise the visibility and keen interest that comes with any allegation relating to an individual of this seniority, it is important for all those involved for the investigative process to be allowed to progress with diligence, proportionality and confidentiality.
"In due course, and subject to appropriate dialogue with the parties at the time, SPA would take the view that it is in the public interest that there should then be a public communication on the outcomes of this process."
The investigation is the latest in a string of controversies to hit the single police force since it was established in 2013.
Holyrood justice committee convener Margaret Mitchell, a Conservative MSP, said the police service was "rudderless".
She said: "The chief constable is under investigation, the chair of the Scottish Police Authority is resigning. The service is rudderless.
"[Justice Secretary] Michael Matheson needs to take control of the situation. This shambolic situation can't be allowed to go on any longer."
A Scottish government spokeswoman said: "We note the Pirc investigation and that they will provide a report to the Scottish Police Authority.
"It would not be appropriate to comment on any current investigation."