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Charlie Sheringham had a penalty saved as Ebbsfleet were frustrated by Eastbourne on Tuesday night.
Sports keeper Lewis Carey kept out Sheringham's spot-kick midway through the second half on a night of few clear-cut chances at the PHB Stadium.
This was a job well done by Eastbourne, who never allowed the Fleet to get into their rhythm.
Ebbsfleet's starting line-up was again unchanged following their 3-0 win at Farnborough on Saturday. Eastbourne, who beat Staines 4-2 at the weekend, brought in Dean Sinclair and Richard Pacquette for Ryan Worrall and Simon Johnson.
Only two points separated the sides at kick-off and there was nothing between them in a tight first half.
Matt Johnson played a good early ball over the top, looking for Sheringham, but his pass had just too much pace on it and Carey got there first.
When Daryl McMahon tried his luck with a low shot from the edge of the box, it looked like being a routine save for Carey but he made rather a meal of things and shovelled the ball behind for a corner.
There were good physical battles all over the pitch, not least between Fleet centre-backs Anthony Acheampong and Kenny Clark and Eastbourne strikers Pacquette and Elliott Charles.
One ball bounced kindly for Pacquette, 30 yards out, and he smashed a shot just wide of the left-hand post.
Gradually, the game started to come to life.
McMahon's free-kick from the left was glanced wide by the head of Acheampong and after the Fleet had survived a goalmouth scramble caused by McCallum's right-wing corner, Eastbourne threw bodies in front of two goalbound McMahon efforts.
Brandon Hall twice had to dive at the feet of visiting attackers in the latter part of the first half and Matt Aldred had a shot deflected over for Eastbourne, but the scoreboard was still blank at the interval.
Hall was back in action at the start of the second half, getting down well to keep out an Eastbourne header when McCallum's inswinging free-kick dropped dangerously at the far post.
Sheringham then met Anthony Cook's deep cross from the left with a technically brilliant volley which struck the inside of the far post - but the assistant's flag had gone up for offside.
On went the arm wrestle of a contest, with the bumper midweek crowd watching in hushed tones.
Charles craned his neck to meet a left-wing cross but Hall caught his header easily.
A cross-shot from the Eastbourne right by full-back Marvin Hamilton took a nick off Aiden Palmer but the deflection wasn't severe enough to wrong-foot Hall, who saved again.
Cook then sped past Kiran Khinda-John down the Fleet left and cut back on himself before whipping a superb low cross into the six-yard box. Billy Bricknell got a touch but Carey made a crucial save from point-blank range.
Bricknell had another shot saved from the edge of the box - and then came the penalty incident.
Cook was clearly and clumsily fouled by Khinda-John after tricking his way to the byline, but Sheringham's spot-kick was palmed out by Carey.
Frustration started to creep into the Fleet's play and the visitors then started to wind up Cook. First it was Sports manager Tommy Widdrington chipping away at the left-winger before former Fleet defender Ian Simpemba started having a go. They succeeded in riling Cook, who was told to calm down by his manager Steve Brown.
Sheringham hooked a half-volley across the face of goal and Clark then sent a looping header from McMahon's right-wing corner onto the roof of the net.
At the other end, McCallum's mis-hit cross floated over Hall and almost dropped in.
McMahon drilled a free-kick straight at Carey, substitute Matty Godden had a strike brilliantly blocked and Cook shot wide in stoppage-time but Eastbourne held on, becoming only the second visiting side so far this season to leave the PHB Stadium with a point.
Ebbsfleet: Hall, Howe, Acheampong, Clark, Palmer, Johnson, Rance, McMahon, Cook, Sheringham, Bricknell (Godden 72mins). Subs not used: Huke, Edwards, Corcoran, Sessegnon.
Eastbourne: Carey, Hamilton, Beale, Aldred, Simpemba, Watts, McCallum, Sinclair (Lovett 88mins), Pacquette (Lok 66mins), Khinda-John (Foot 83mins), Charles. Subs not used: Worrall, Taylor.
Attendance: 944.