LOCK-UP AGREEMENT Madison Air Solutions Corporation Lock-Up Agreement April 15, 2026

Exhibit 10.5

LOCK-UP AGREEMENT

Madison Air Solutions Corporation

Lock-Up Agreement

April 15, 2026

Madison Air Solutions Corporation

▇▇▇ ▇. ▇▇▇▇ ▇▇▇▇▇▇, ▇▇▇▇▇ ▇▇▇▇

Chicago, IL 60606

Attn: General Counsel

Re: Madison Air Solutions Corporation - Lock-Up Agreement

Ladies and Gentlemen:

The undersigned understands that Madison Air Solutions Corporation, a Delaware corporation (the “Company”), is contemplating effecting a public offering (the “Public Offering”) of shares (the “Shares”) of the Class A common stock, par value $0.0000001 per share, of the Company pursuant to a Registration Statement on Form S-1 (the “Registration Statement”) filed with the U.S. Securities and Exchange Commission (the “SEC”). As used herein, the term “Common Stock” refers to shares of the Company’s Class B Common Stock, par value $0.0000001 per share owned by the undersigned on the date hereof and any securities issued by the Company in connection therewith, whether upon conversion to shares of the Company’s Class A Common Stock, par value $0.0000001 per share, as a dividend, in a stock split or otherwise.

In consideration of the undertaking by the Company to offer and sell Shares in the Public Offering, and of other good and valuable consideration the receipt and sufficiency of which is hereby acknowledged, the undersigned agrees that, during the period beginning from the date of this Lock-Up Agreement and continuing to and including the date two (2) years after the date of the final prospectus relating to the Public Offering (the “Prospectus”) (such period, the “Lock-Up Period”), the undersigned shall not (i) offer, sell, contract to sell, pledge, grant any option, right or warrant to purchase, purchase any option or contract to sell, lend or otherwise transfer or dispose of any shares of Common Stock, or any options or warrants to purchase any shares of Common Stock, or any securities convertible into, exchangeable for or that represent the right to receive shares of Common Stock (such shares of Common Stock, options, rights, warrants or other securities, collectively, “Lock-Up Securities”), including, without limitation, any such Lock-Up Securities currently beneficially owned (as such term is used in Rule 13d-3 of the Securities Exchange Act of 1934, as amended (the “Exchange Act”), (ii) engage in any hedging or other transaction or arrangement (including, without limitation, any short sale or the purchase or sale of, or entry into, any put or call option, or combination thereof, forward, swap or any other derivative transaction or instrument, however described or defined) which is designed to or which reasonably could be expected to lead to or result in a sale, loan, pledge or other disposition (whether by the undersigned or someone other than the undersigned), or transfer of any of the economic consequences of ownership, in whole or in part, directly or indirectly, of any Lock-Up Securities, whether any such transaction or arrangement (or instrument provided for thereunder) would be settled by delivery of Common Stock or other securities, in cash or otherwise (any such sale, loan, pledge or other disposition, or transfer of economic consequences, a “Transfer”), (iii) make any demand for or exercise any right with respect to the registration of any Lock-Up

Securities or (iv) otherwise publicly announce any intention to engage in or cause any action, activity, transaction or arrangement described in clause (i), (ii) or (iii) above. The undersigned represents and warrants that the undersigned is currently a party to any agreement or arrangement that provides for, is designed to, or reasonably could be expected to lead to, or result in, any Transfer during the Lock-Up Period.

Notwithstanding anything set forth herein to the contrary, the undersigned may otherwise Transfer Lock-Up Securities (a) pursuant to a bona fide third-party tender offer, merger, consolidation or other similar transaction that is approved by the Board of Directors of the Company and made to all holders of the Company’s capital stock involving a Change of Control of the Company (for purposes hereof, “Change of Control” shall mean the transfer (whether by tender offer, merger, consolidation or other similar transaction), in one transaction or a series of related transactions, to a person or group of affiliated persons, of shares of capital stock if, after such transfer, such person or group of affiliated persons would hold at least a majority of the outstanding voting securities of the Company (or the surviving entity)); provided that in the event that such tender offer, merger, consolidation or other similar transaction is not completed, the undersigned’s Lock-Up Securities shall remain subject to the provisions of this Lock-Up Agreement, or (b) to an affiliate or a member of the undersigned if such transferee executes a Lock-Up Agreement in the same form as this Lock-Up Agreement, which Lock-Up Agreement would terminate at the end of the Lock-Up Period.

The undersigned now has, and, except as contemplated by the foregoing paragraph, for the duration of this Lock-Up Agreement (subject to the transfers permitted by the immediately preceding paragraph) will have, good and marketable title to the undersigned’s Lock-Up Securities, free and clear of all liens, encumbrances and claims whatsoever. The undersigned also agrees and consents to the entry of stop transfer instructions with the Company’s transfer agent and registrar against the transfer of the undersigned’s Lock-Up Securities (subject to the transfers permitted by the immediately preceding paragraph).

The undersigned acknowledges and agrees that the Company has not made any recommendation or provided any investment or other advice to the undersigned with respect to this Lock-Up Agreement or the subject matter hereof, and the undersigned has consulted its own legal, accounting, financial, regulatory, tax and other advisors with respect to this Lock-Up Agreement and the subject matter hereof to the extent the undersigned has deemed appropriate. The undersigned further acknowledges and agrees that the Company has not made and is not making a recommendation to the undersigned to enter into this Lock-Up Agreement or to transfer, sell or dispose of, or to refrain from transferring, selling or disposing of, any shares of Common Stock, and nothing set forth in such disclosures or herein is intended to suggest that the Company is making such a recommendation.

This Lock-Up Agreement shall automatically terminate and the undersigned shall be released from all of its obligations hereunder upon the earlier of (i) the date on which the Registration Statement filed with the SEC with respect to the Public Offering is withdrawn, (ii) the date on which the Company notifies the undersigned in writing that it does not intend to proceed with the Public Offering and (iii) December 31, 2026, in the event that the Registration Statement filed with the SEC with respect to the Public Offering has not been declared effective; provided, however, that the Company may, by written notice to the undersigned prior to such date, extend such date by a period of up to an additional 90 days. The undersigned acknowledges and agrees that this Lock-Up Agreement is in addition to, and not in lieu of, any lock-up agreement entered into by the undersigned with the underwriters in connection with the Public Offering, and the

restrictions and obligations set forth herein shall be separate from and in addition to any restrictions and obligations set forth in any such lock-up agreement with the underwriters.

The undersigned understands that the Company is relying upon this Lock-Up Agreement in proceeding toward consummation of the Public Offering. The undersigned further understands that this Lock-Up Agreement is irrevocable and shall be binding upon the undersigned’s heirs, legal representatives, successors and assigns. The undersigned hereby represents and warrants that the undersigned has full power and authority to enter into this Lock-Up Agreement. This Lock-Up Agreement shall be governed by, and construed in accordance with, the laws of the State of New York, without regard to principles of conflict of laws that would result in the application of any law other than the laws of the State of New York. This Lock-Up Agreement may be delivered via facsimile, electronic mail (including pdf or any electronic signature complying with the U.S. federal ESIGN Act of 2000, e.g., ▇▇▇.▇▇▇▇▇▇▇▇.▇▇▇ or ▇▇▇.▇▇▇▇▇▇▇▇.▇▇▇) or other transmission method, and any counterpart so delivered shall be deemed to have been duly and validly delivered and be valid and effective for all purposes.

[Signature Page Follows]

Very truly yours,

	MADISON INDUSTRIES HOLDINGS LLC
	By:	/s/ ▇▇▇▇▇ ▇▇▇▇
		(duly authorized signature)
	Name:	▇▇▇▇▇ ▇▇▇▇
		(please print full name)
	Title:	President and Chief Executive Officer
		(please print full title)

[Signature Page to Lock-Up Agreement]