The clarify of the question was the wrong statement about the presence of the US forces in Syria because i have no confirmation from official sources on what grounds the US presence correlates with the UN Charter's Article 51 and which article of this UN Charter justifies it today.
So, the claim that intervention can be based on article 51 looks doubtful. But if it is true official version it would like to see a competent confirmation of this version.
But i found "the judge-head" of the UN - it is the the International Court (ICJ)
And the ICJ have the advisory opinions to the cases and the UN Charter articles.
So for this moment im considering as a similar precedentt the following:
On 13 July 2004 ICJ Advisory opinion on the Legal Consequences of the Construction of a Wall in the OPT
Here-5:
Article 51 of the Charter . . . recognizes the existence of an inherent right of self-defence in the case of armed attack by one State against another State. However, Israel does not claim that the attacks against it are imputable to a foreign State.
The Court also notes that Israel exercises control in the Occupied Palestinian Territory and that, as Israel itself states, the threat which it regards as justifying the construction of the wall originates within, and not outside, that territory. The situation is thus different from that contemplated by Security Council resolutions 1368 (2001) and 1373 (2001), and therefore Israel could not in any event invoke those resolutions in support of its claim to be exercising a right of self-defence.
Consequently, the Court concludes that Article 51 of the Charter has no relevance in this case.” (Para. 139.)
And here-6:
Article 51 of the Charter provides that “Nothing in the present Charter shall impair the inherent right of individual or collective self-defence if(!) an armed attack occurs against a Member of the United Nations . . .” Moreover, in the resolutions cited by the Court, the Security Council has made clear that “international terrorism constitutes a threat to international peace and security”
And here-6 too precedent:
In its resolution 1368 (2001), adopted only one day after the September 11, 2001 attacks on the United States, the Security Council invokes the right of self-defence in calling on the international community to combat terrorism. In neither of these resolutions did the Security Council limit their application to terrorist attacks by State actors only, nor was an assumption to that effect implicit in these resolutions. In fact, the contrary appears to have been the case. (See Thomas Franck, “Terrorism and the Right of Self-Defense”, American Journal of International Law, Vol. 95, 2001, pp. 839-840.)
"In the aftermath of the attacks, many U.S. citizens believed that the attacks had "changed the world forever." The Bush administration announced a war against terrorism, with the goal of bringing Osama bin Laden and al-Qaeda to justice and preventing the emergence of other terrorist networks." Wiki
11/11 only because => al-Qaeda is a legitimate target by the UN Charter 51!!
And here-9(probably if US citizens are located there):
Paragraph 6 of Article 49 of the Fourth Geneva Convention also does not admit for exceptions on grounds of military or security exigencies. It provides that “the Occupying Power shall not deport or transfer parts of its own civilian population into the territory it occupies”.
So, my clarify is not actual for now -
I understand that the US base their actions on article 51, but do they also violate 2.4 or not? Is Art. 51 a legal exemption to avoid Art. 2.4? Are both contradictory and a thus a "loophole" in International Law? How are both to be interpreted together?
Because, i am not understanding why the US did it by the UN Charter logic and i can't find an official version of the US why they do it now.
But the other questions are still actual, i don't have answers on them.
But i understand that:
ICJ said that 51 is apply to 2.4 but in "if an armed attack occurs against a Member of the United Nations" reason
But i don't know, have the US claimed that ISIL had attack occurs them?
Argumentation about "unable or unwilling" doctrine or "Legally speaking 51 is an exemption from 2.4" does not looks sufficient for use the Article 51, because it creats contradictory between 51 and 2.4. And the source Just Security, that claimed "Legally speaking 51 is an exemption from 2.4" - doesn't look as does not look authorized because it is non-government or the UN institution. It is good as consultation opinion, but it have no a competence status to be a judgement base in the question.
But the problem, i have no information with an official reason of the US version - at start, and at today too.
The answer:
At the UN Charter logic set out in ICJ interpretation there is no contradictory between 51 and 2.4.
Also i have no information about the US official claims about ISIL attack which could be the basis for the Artical 51 usage. But They said "because 51" as i know. As i understand that was not "if an armed attack occurs" by ISIL.
The contradictory appears by the "unable or unwilling" doctrine(officially) and/or "Legally speaking 51 is an exemption from 2.4"(don't know officially status) owned apparently to the US side. This claims are cheating and erasing the sense of "if an armed attack occurs" (2). Therefore it needs "Legally speaking 51 is an exemption from 2.4". But i don't know is it official justification or not. But also it can't be considered the UN official interpretation.
i have no information does the 3 usable to justify intervention in the US in Syria and is this a today excuse to stay there now.