As outlined in the Constitution, the president of the United States and the U.S. Senate do not have shared powers. Under the Constitution, however, both the president and the Senate are allowed the means to check and balance each others' power.
The Senate's constitutional role of advice and consent gives it confirmation authority over presidential appointments. This includes ambassadors, Supreme Court justices and cabinet secretaries. The Senate also has the power to try impeachment cases of federal officials, including the president of the United States. In conjunction with the House of Representatives, the Senate can also override a presidential veto if both houses of Congress muster a two-thirds vote to do so within its respective chambers. As the nation's chief diplomat, the president has the ability to negotiate and sign treaties, but these must ultimately be ratified by the Senate.
Conversely, the president of the United States can veto bills sent to him by Congress. This means that bills that originate in the Senate and pass through the House of Representatives can ultimately be thwarted by the President if he refuses to sign them into law by issuing a veto statement. Furthermore, when signing a bill, the president has the ability to issue a "signing statement," which allows him to express his opinion on the constitutionality of the bill in question. These statements serve to guide the manner in which the law is implemented by executive agencies. Also, if the Senate (along with the House of Representatives) adjourns within 10 days of sending a bill to the president, the president may choose not to act on the bill, which effectively blocks it from becoming law. This is known as a pocket veto, which cannot be overridden by Congress.