Fianna Fáil TD and Chairman of the Oireachtas finance committee John McGuinness has said the Central Bank has all the answers on Davy stockbrokers but it is restricted by legislation as to what it can say to the committee.
Speaking on RTÉ's This Week, Mr McGuinness said it is still important the Central Bank comes before the committee to be questioned and to secure as "much information as possible".
On Tuesday, the Central Bank fined Davy €4.13m for breaching market rules in relation to a transaction in November 2014 involving 16 of the broker's own staff.
The fine was the biggest of its kind ever levied on a broker in Ireland.
The board of the company also accepted the resignations of Kyran McLaughlin from his role as a non-executive Director and of Barry Nangle from his executive role as Head of Bonds at Davy.
The company said all three offered to step down following the recent fine by the Central Bank.
Today, Mr McGuinness appealed for Davy executives to come before the committee and explain what happened, and what they are going to do to for the future and to "rebuild trust that has been woefully damaged".
"We would like to know how this occurred, how they were able to get away with it so easy.
"If Davy's is to rebuild trust they need to give titles of individuals concerned."
He said the Minister in turn needs to require legislation to be changed to make the investigation more transparent and public.
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"Essentially in terms of Davy they need to tell us they are absolutely sure this didn't happen in any other case. I think this is essential for clients of the firm they know that."
He added that it is important the Central Bank looks at the company in more detail to see if more incidents like this happened.
"We should not let the story lie at this point, we need to know the full truth as it reflects the image of the country and company," he added.
Minister for Public Expenditure Michael McGrath described what happened at Davy as an extremely serious breach of trust.
He said serious questions have been raised about how the company dealt with the Central Bank and how it handled matters this week when the news emerged.
He said the National Treasury Management Agency (NTMA), which manages the national debt and borrows on behalf of the State, is a client of Davy and it has written to the brokers to set out its extremely serious concerns in relation to the values and behaviours demonstrated by the regulations' breach.
He said the NTMA will advise Government about the response it gets.
He said the Government and the NTMA regard this as an unacceptable breach of trust.
Speaking on RTÉ's The Week in Politics, Sinn Féin finance spokesperson Pearse Doherty said the State should first suspend its relationship with Davy's and failure to do so gives Davy's reputational cover.
While three people resigned there are still 13 others who have not and the State should not deal with the company until there is a thorough review and until it is clear what individual accountability will apply to those involved, he added.
He said the Central Bank does have powers now to investigate individuals and Sinn Féin wants to know that is happening and that a wider investigation is taking place in Davy's, to see if this has happened with any other transactions and if any files have been referred to the gardaí.
Speaking on the same programme, Labour finance spokesperson Ged Nash said that an internal investigation in Davy's is not good enough.
Individual responsibility, accountability and liability is at the core of this, he commented.
Mr Nash added that the section of law used by the Central Bank to investigate the transaction can also be used to investigate the conduct of individuals.
He said we must hear from the Central Bank where that is at. He said Davy's offering three senior resignations does not address the problems.